Doctors have reiterated their support for overhauling the organ donation system.
The Human Transplantation Bill, which could become law by 2015, would mean adults living in Wales would have to "opt-out" of the organ donation system if they did not want to donate after death.
Ms Griffiths told the health committee that, on average, three people a month in Wales die while waiting for an organ transplant and a new "soft opt-out" organ donation system would "significantly improve" on that situation.
BBC: Organ donation: Opt-out bill to exclude some transplants
Officials say the country would instead rely on a new national donation system for organ transplants.
If that bill is passed, an opt-out system for organ donation could become law around the beginning of 2015.
In Wales the situation with regards to adopting an opt out system for organ donation is a lot more advanced than in England.
The Welsh Government's just released two documents as part of its consultation on legislation to bring in a soft opt out system of organ donation.
If the legal advice is that the proposals do cross into areas of individual liberty and human rights, then it is possible that it would be determined that a system of organ donation cannot be devolved.
The Welsh Government has commissioned Beaufort Research to carry out a qualitative survey of people's attitudes to an opt-out system of organ donation which comes to some striking - and perhaps counterintuitive - conclusions about this very issue.
Huang Jiefu, vice minister of health, was quoted by Xinhua as saying that a trial system for public organ donation has been launched in some areas.
The Nuffield Council's report also considers moving to an "opt-out" system for deceased organ donation, under which it would be assumed that people consent to donation unless they either opt out before they die, or their family later objects.
Mrs Taylor backs the Welsh government's plans change organ donation in Wales to an "opt-out" system.
The BMA in Scotland renewed calls for an "opt-out" system, arguing that 90% of the population support organ donation, yet only 40% have signed up.
In 2008, the Organ Donation Taskforce said a 50% increase in organ donation was achievable in the UK within five years, but it stopped short of recommending an opt-out system.
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