It says an "overwhelming majority" of the public now supports a change in the law to allow assisted dying, and that refusing to debate the issue in Parliament would be "heartless".
It also claims one Briton goes abroad to end their own life every two weeks.
Other signatories include the Booker Prize-winning author Ian McEwan, whilst notable backers of the campaign include actors Hugh Grant, Sir Patrick Stewart and Zoe Wannamaker.
Prime Minister David Cameron has said MPs will be given a free vote if the issue is debated in Parliament.
The Falconer Bill, which is currently going through the House of Lords, would allow doctors to help terminally-ill people take their own life, but only if they had less than six months to live and had demonstrated a "clear and settled intention" to end their lives.
The Bill does not have enough time to become law before next year's general election in May.
It's not the first time Lord Carey has given his backing to assisted dying.
Writing in the Daily Mail in July he announced he had changed his mind on the issue and would no longer 'fiercely oppose' the move.
"I would have used the time-honoured argument that we should be devoting ourselves to care, not killing.
"I would have paraded all the usual concerns about the risks of 'slippery slopes' and 'state-sponsored euthanasia'.
"But those arguments which persuaded me in the past seem to lack power and authority when confronted with the experiences of those approaching a painful death.
"It fails to address the fundamental question as to why we should force terminally ill patients to an unbearable point. It is the magnitude of suffering that has been preying on my mind as the discussion over the right to die has intensified.
"Today we face a central paradox. In strictly observing the sanctity of life, the Church could now actually be promoting anguish and pain, the very opposite of a Christian message of hope."