Health and Social Services Minister Lesley Griffiths introduced the human transplantation bill to the assembly on 4 December 2012.
If the measure becomes law, as is planned for 2015, consent for the donation of organs will be deemed given unless the deceased had expressly wished otherwise during his or her lifetime.
The health minister said that the measure is intended to increase the number of organs available for patients and encourage debate on the issue of donation.
She said: "Forty one people in Wales died last year waiting for organ transplant.
"Organ transplant is one of the most successful medical interventions."
Government officials claim that a change in the law could lead to 15 extra donors leaving around 45 more organs for transplant every year.
Conservative Health Spokesperson Darren Millar confirmed that his party supported the measure in principle, but emphasised the fact that the presumed consent system wouldn't increase donation rate on its own.
"Sweden has a lower donation rate than the UK and Spain attributes its high donation rates to processes that function on the back of legislation," he argued.
He also raised concerns about whether the legislation would be compatible with Muslim and Christian groups, and asked whether the health minister would consider a presumed opt-out for these communities.
Ms Griffiths didn't accept Mr Millar's concerns, claiming that a member of the Muslim community had told her that a "load of rubbish" was being talked about organ donation within his community.
Liberal Democrat Health Spokesperson Kirsty Williams and Plaid Cymru Health Spokesperson Elin Jones expressed their support for the measure, but both warned that infrastructural work also needed to be done to ensure that the measure succeeded in its stated intention to increase the amount or organs available for transplant.
Monmouth AM Nick Ramsay expressed concern at the fact that families of the deceased would not be given a veto over consent.
In response, Ms Griffiths gave her assurances that due weight would be given to the wishes of the family.
AMs also considered a draft order relating to water supply.
The order is intended to allow inspectors to recover the cost of work undertaken in relation to water supplies from water suppliers and was passed unanimously.