They must be expected to die within six months, according to two independent doctors, and have the request signed off by a High Court judge.
The bill has been the focus of fierce and passionate disagreement at Westminster and throughout the country over the past six weeks, with Keir Starmer’s Cabinet split on the issue.
Health Secretary Wes Streeting and Justice Secretary Shabana Mahmood are among those who opposed the move, while Home Secretary Yvette Cooper and Work and Pensions Secretary Liz Kendall supported the bill.
The bill is now due to pass to committee stage to be examined in detail.
There is strong, bipartisan support for assisted dying for the terminally ill, according to polls carried out by YouGov.
A study by the same firm last week showed that 73 percent of Britons think assisted dying should be legal. However, 19 percent said that while they support assisted dying in principle, they oppose it in practice because they don’t believe adequate laws can be created to regulate it.
Reacting to the result, Leadbeater told the BBC she felt “overwhelmed” but added that the debate had shown parliament in its best light and the legislative process to follow would be “very thorough.”