Wales could become the first part of the UK to introduce a policy of assumed consent for organ donation, despite reservations by religious leaders.
The Welsh government wants to introduce a scheme in which people would be presumed to have given consent to have their organs and organic tissue harvested after they die but would allow for a specific opt-out policy. Under present British law, organ donation is an opt-in procedure.
The Human Transplantation (Wales) Bill would allow relatives of the deceased to stop their organs being donated.
Ministers hope that the scheme would increase the number of donors for transplants by at least a quarter.
Roy Thomas, chief executive of the Kidney Wales Foundation (KWF), said: "The UK is one of the lowest donor rate countries in Europe.
"The new bill is progressive law. We are proud to have led the debate in favour of this law.
"When introduced in Belgium only 2% opted out. Currently, only around a third of the Welsh population is on the organ donor register and this is around the same for the UK as a whole."
He added: "Kidney Wales believes the presence of the family is essential - both as a source of necessary information about the potential donor and in order to ensure that donation does not go ahead in the face of the deceased's known objection to organ donation."
A group of faith leaders, including the Archbishop of Wales, Dr Barry Morgan, want a "soft opt-out option" ithat would take into consideration the views of bereaved families.
The signatories, who include the Catholic Archbishop of Cardiff, George Stack, said: "The law must be amended in this way if it is not to be inhuman, unfeeling before the suffering of relatives, and a danger to the public trust and support which are necessary for the practice of organ donation to flourish.
"Without this there is a real danger that the law could backfire."
However, Melanie Wager, who received a kidney in July 2010, has welcomed the proposed changes to the law.
She said: "Waiting for an organ is an extremely difficult time for anyone - it is like being on death row and it seems as if you are being further punished for being ill.
"Mentally, it is cruel for the patient and the caring family."
Welsh health minister Mark Drakeford added that majority of Welsh people would want to be a donor.
He said: "Deemed consent will bring about a cultural shift in the way donation proceeds in Wales.
"It will alter the nature of some of the most difficult conversations that any family might face, and it will help to ensure that the wishes of that substantial majority of Welsh citizens who say, in survey after survey, that they would wish to be a donor, are put into practice in those very rare and special circumstances when donation is possible."
If the bill were passed, the presumed consent system could be implemented by 2015.
Figures show one person dies every week in Wales waiting for an organ transplant. In the UK as a whole the figure is one every three days.