The assembly government warned the powers will take time to achieve
Wales could become the first part of the UK introduce an opt-out system of organ donation under plans by the assembly government.
It would mean that Welsh residents would be presumed to be organ donors unless they have joined an opt out register or immediate relatives object.
Other exclusions include if a person's identity or place of residence cannot be confirmed.
But there has been criticism of the plans by the Conservatives.
The move is aimed at boosting organs available for transplants.
The so called 'soft opt-out' system is similar to that already used in Belgium and Portugal, where organ donation rates are far higher than in the UK.
During the assembly government's consultation the British Medical Association (BMA) had called for an opt out system.
Jonathan Morgan explains his opposition to presumed consent
It said an estimated 30 people died waiting for organ transplants in Wales in a year.
In a written cabinet statement released on Friday, Health Minister Edwina Hart said: "I have made clear that I am personally in favour of presumed consent as a way of increasing organ donation.
"However, I recognise that this is a sensitive and emotive subject.
"That is why I have wanted to spend time ensuring we gauged people's views before coming to a conclusion."
We hope the rest of the UK will now follow the lead that Wales has given
Roy J Thomas, Kidney Wales Foundation
She said during public consultation she had specifically asked the public if the assembly government should move forward with a proposed legislative change in Wales.
"The majority of responses supported a change to the organ donation system in Wales to a soft opt-out system," she added.
"Cabinet colleagues have agreed with my proposal to explore the possibility of introducing a soft opt-out system for organ donation in Wales and that a bid for a Legislative Competence Order [a so-called Welsh law] be submitted.
"This will take time to achieve so we must continue to do all we can to raise awareness of this issue and encourage people to sign up to the organ donation register."
The move has also won the backing of the Kidney Wales Foundation, which warned that someone waiting for a transplant in Wales died every 11 days.
The charity's chairman, Roy J Thomas said: "We know the overwhelming majority of people in Wales would like their organs to be donated after death but only 28% of people are on the NHS organ donor register.
"Cruelly, this means even more people are dying and waiting.
"This decision by the minister will break new ground in the UK.
"Many other countries in Europe operate a system of opt-out, and it is a large part of why they have far higher organ donation rates.
"We hope the rest of the UK will now follow the lead that Wales has given."
However Jonathan Morgan, Conservative AM and a former chair of the assembly health committee, said he disagreed with the proposals.
"I think it is far more preferable for someone to want to opt-in to something than for the government to presume you have no objection because you have not made any utterance about it," he said.
"I think that is the wrong way to go about it.
"I don't think it is acceptable for the government to make a presumption about what your beliefs and views are," he added.
But the Liberal Democrats have welcomed the plans.
Peter Black, the party's health spokesman said:"I am very pleased that the minister has decided to bid for the power to make this change to the organ donation rules in Wales. This is a policy that the Welsh Liberal Democrats fully support and is greatly needed in Wales.
"The soft-opt out system is the best way of increasing the number of donations that are made, while giving families and individuals the safeguards against having their organs removed against their will."
The Head of Science and Ethics for the BMA, Dr Vivienne Nathanson, said the organisation was "delighted" with the moves being made by the assembly government.
"We hope that the debate in Wales stimulates similar discussions in the rest of the UK.
"The BMA still believes that one of the best ways to increase organ donation is to introduce presumed consent with safeguards."
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