Assisted dying generally refers to a person who is terminally ill receiving lethal drugs from a medical practitioner, which they administer themselves.
It is banned in England, Wales and Northern Ireland, with a maximum prison sentence of 14 years.
Under the bill, adults expected to die within six months could seek help to end their life.
Two doctors and a High Court judge would need to agree they were eligible.
Labour MP Kim Leadbeater, who is behind the bill, said it could prevent “distressing” deaths.
Dame Esther said she told Ms Leadbeater the current laws in the UK were “terrible” and “cruel” because they forced people to travel to Switzerland alone to die.
She said if her family went with her, they risked being prosecuted for assisting her death.
“I would much rather be able to die in my own home surrounded by people who are close to me,” she said.
“But that choice won’t be open to me, because even if we get a ‘yes’ vote in this debate… you can’t get through the next stage in under a year.
“And my miracle drug is working jolly hard but I can’t expect it to keep me alive for a year.”