The broadcaster has joined Swiss clinic Dignitas, which lets people have an assisted death, but her family could currently be prosecuted if they were to travel there with her.
Dame Esther told the BBC it was “important that the law catches up with what the country wants”.
Assisted dying is banned in England, Wales and Northern Ireland, with those convicted facing up to 14 years in jail.
In Scotland, it is not a specific criminal offence, but assisting the death of someone can leave a person open to murder or other charges.
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7:10
What is assisted dying?
Starmer: Free vote ‘seems appropriate’
Sir Keir voted in favour of legislating for assisted dying in 2015, when a private members bill was brought to the Commons by Labour MP Rob Marris.
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Members were given a free vote on the issue – meaning their political parties did not pressure them to vote in a particular way – and they overwhelmingly rejected a change in the law by 330 votes to 118.
Sir Keir said while there are “obviously strong views both ways on this, which I respect”, another private member’s bill and free vote “seems appropriate”.
Both Housing Secretary Michael Gove and Work and Pensions Secretary Mel Stride have also said they would be willing to see a fresh parliamentary debate on the issue.
The prime minister’s official spokesman said the government’s position has not changed, so it remains a matter for parliament to decide and “an issue of conscience for individual parliamentarians rather than government policy”.
But what about his style ‘prince’? Some want that ditched too.
It’s a complicated but not impossible process. Andrew could, of course, just stop using it voluntarily.
Some want him to give up his home, too. For a non-working royal, the stately Royal Lodge, with its plum position on the Windsor Estate, is an uncomfortable optic.
With the reputation of the monarchy at risk, William does not want to appear weak. He’s putting loyalty to “the firm” firmly above his familial relationships.
Prince Andrew has always strongly denied the allegations, and restated on Friday: “I vigorously deny the accusations against me”. Sky News has approached him for comment on the fresh allegations set out in the Mail on Sunday.
But with Virginia Giuffre’s tragic death and posthumous memoir due out on Tuesday, Buckingham Palace will be braced for more scandal.
When Andrew gave up his titles, there was certainly a sense of relief.
There is now a sense of dread over what else could emerge.
Sky News’ royal commentator has explained why Prince Andrew has not given up being called a prince – while another expert has said “the decent thing” for him to do would be “go into exile” overseas.
Andrew announced on Friday that he would stop using his Duke of York title and relinquish all other honours, including his role as a Royal Knight Companion of the Most Noble Order of the Garter.
However, he will continue to be known as a prince.
Royal commentator Alastair Bruce said that while Andrew’s other honours and titles were conferred to him later in life, he became a prince when he was born to Elizabeth II while she was queen.
He told presenter Kamali Melbourne: “I think […] that style was quite special to the late Queen,” he said. “And perhaps the King, for the moment, thinks that can be left alone.
“It’s a matter really for the King, for the royal household, perhaps with the guidance and advice of government, which I’m sure they are taking.”
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2:49
Who pushed Andrew to drop his titles?
Since Andrew’s announcement, there has been speculation over whether any further measures will be taken – and one author has now called for him to “go into exile”.
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Andrew Lownie, author of The Rise And Fall Of The House Of York, said: “The only way the story will go away is if he leaves Royal Lodge, goes into exile abroad with his ex-wife, and is basically stripped of all his honours, including Prince Andrew.”
Royal Lodge is the Windsor mansion Andrew lives in with his ex-wife, Sarah Ferguson, who has also lost her Duchess of York title.
Image: Andrew and his former wife continue to live on the Windsor estate. Pic: Reuters
Mr Lownie continued: “He makes out he’s an honourable man and he’s putting country and family first. Well, if he is, then the optics look terrible for the monarchy. A non-working royal in a 30-room Crown Estate property with a peppercorn rent.
“He should do the decent thing and go. And frankly, he should go into exile.”
Mr Lownie added if the Royal Family “genuinely want to cut links, they have to put pressure on him to voluntarily get out”.
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1:11
Windsor’s take on Prince Andrew
Andrew’s decision to stop using his titles was announced amid renewed scrutiny of his relationship with paedophile Jeffrey Epstein, and fresh stories linked to the late Virginia Giuffre.
Ms Giuffre, who was trafficked by Epstein, alleged she was sexually assaulted by Andrew on three occasions – which he has always vigorously denied.
Bereaved families whose loved ones took their own lives after buying the same poison online have written to the prime minister demanding urgent action.
Warning: This article contains references to suicide
The group claims there have been “multiple missed opportunities” to shut down online forums that promote suicide and dangerous substances.
They warn that over 100 people have died after purchasing a particular poison in the last 10 years.
Among those who have written to Downing Street is Pete Aitken, whose daughter Hannah was 22 when she took her own life after buying the poison from a website.
Hannah was autistic and had ADHD. She was treated in six different mental health hospitals over a four-year period.
He said: “Autistic people seem to be most vulnerable to this kind of sort of poison and, you know, wanting to take their lives.”
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4:05
Pete Aitken speaking to Sky News
Sky News is not naming the poison, but Hannah was able to buy a kilogram of it online. Just one gram is potentially fatal.
“There’s this disparity between the concentration required for its legitimate use and that required for ending your life. And it seems quite clear you could make a distinction,” Mr Aitken said.
Analysis from the Molly Rose Foundation and the group Families and Survivors to Prevent Online Suicide Harms says at least 133 people have died because of the poison. It also says coroners have written warnings about the substance on 65 separate occasions.
The report accuses the Home Office of failing to strengthen the regulation of the poison and says not enough is being done to close dangerous suicide forums online.
Lawyers representing the group want a public inquiry into the deaths.
In a joint letter to the prime minister, the families said: “We write as families whose loved ones were let down by a state that was too slow to respond to the threat.
“This series of failings requires a statutory response, not just to understand why our loved ones died but also to prevent more lives being lost in a similar way.”
The group’s lawyer, Merry Varney, from Leigh Day, said: “The government is rightly committed to preventing deaths through suicide, yet despite repeated warnings of the risks posed by an easily accessible substance, fatal in small quantities and essentially advertised on online forums, no meaningful steps have been taken.”
Image: Hannah’s dad is one of the family members to have signed the letter
A government spokesperson said: “Suicide devastates families and we are unequivocal about the responsibilities online services have to keep people safe on their platforms.
“Under the Online Safety Act, services must take action to prevent users from accessing illegal suicide and self-harm content and ensure children are protected from harmful content that promotes it.
“If they fail to do so, they can expect to face robust enforcement, including substantial fines.”
They added that the position is “closely monitored and reportable under the Poisons Act, meaning retailers must alert authorities if they suspect it is being bought to cause harm”.
“We will continue to keep dangerous substances under review to ensure the right safeguards are in place,” they said.
Anyone feeling emotionally distressed or suicidal can call Samaritans for help on 116 123 or email jo@samaritans.org in the UK. In the US, call the Samaritans branch in your area or 1 (800) 273-TALK.