Peers were split over the Assisted Dying Bill when it was debated for the first time in the House of Lords on Friday.
Having heard from more than 120 peers during the debate, they agreed to send the Bill to committee stage for further scrutiny.
Lord Falconer's Bill would allow doctors to prescribe lethal drugs to a patient with less than six months to live.
While the Prime Minister has said he is "not convinced" about assisted dying, he has supported the idea of a Commons debate on the issue.
But without government backing such a debate is unlikely to happen, meaning that the Bill will almost certainly not pass before the next election.
Lord Falconer opened the debate saying: "The current situation leaves the rich able to go to Switzerland, the majority reliant on amateur assistants, the compassionate treated like criminals and no safeguards in terms of undue pressure now".
Peers on both sides of the debate spoke from personal experience and testimonies from those who had contacted them.
Lord Dobs said: "I support the Bill, because I do not wish to deny other people something that I might want myself some day in future.
"Some years ago, I went to see a friend of mine who had motor neurone disease. His whole family was there; he could no longer communicate, except on a keyboard, and he had assembled the family because they together wanted to ask me to support a change in the law. He died not long afterwards, but it was a heartfelt wish on his part. How could I say no to such a plea?"
However Baroness Campbell, speaking from her wheelchair with a respiratory aid, condemned the Bill which she said was about her and those like her who suffer from degenerative conditions.
She said: "The Bill purports to offer choice – the option of premature death instead of pain, suffering and disempowerment – but it is a false choice. It is that of the burglar who offers to mug you instead."
She later added: "The Bill has become a runaway train, and the more frightening because of that."
Former Conservative cabinet minister, Lord Tebbit said: "No one could dispute the good intentions of the Bill, but the road to hell is paved with good intentions.
"I noticed that the noble Baroness, Lady Greengross, referred several times to the 'right' that we all have to take our own lives. We do not have that right; we have only the capacity to do it."
The Archbishops of York and Canterbury are also opposed to the Bill.
In the debate the Archbishop of York John Sentamu said: "The present Bill is not about relieving pain or suffering; it makes that quite clear in its definition of a terminally ill patient to include those whose progressive illness can be relieved but not reversed.
"The Bill is about asserting a philosophy [...] – that is, the ancient Stoic philosophy that ending one's life in circumstances of distress is an assertion of human freedom."
The Rt Rev Dr Alan Wilson, the Bishop of Buckingham, and former Archbishop Lord Carey have broken away from the Church of England's official stance and supported the Bill.
Viscount Craigavon welcomed their divergence from what he described as an "apparent Orwellian consensus" in the Church of England.