An earlier idea in the bill was for the Senedd to have a veto over all the important parts of the law – allowing MSs to decide when it comes into force.
But last week MPs decided to take that out.
On Wednesday, James – who is the Welsh government’s most senior legal adviser and a MS herself – said the power was removed to avoid giving Welsh ministers powers that were outside the Senedd’s control.
That is because the Government of Wales Act, external specifically prohibits the Senedd from amending the law on suicide.
But she added: “The bill still contains the key provisions that mean that the NHS or any public body in Wales will not be able to provide voluntary assisted dying services until regulations have been laid by ministers and there has been an affirmative vote in the Senedd to that effect.”
Giffard said it was “regrettable” the “veto” power had been removed.
He told BBC Wales: “While the bill has passed through Westminster, it’s important to remember that the Senedd did not support it when it was last debated.
“A decision of such gravity must not be imposed on Wales without the explicit consent of its elected representatives.”
Last October the Senedd voted against a motion calling for a new law to allow assisted dying – 19 to 26, with nine abstentions.
Health Secretary Jeremy Miles and First Minister Eluned Morgan were among those opposed.
The bill in the UK Parliament was different to October’s motion, with a wider scope proposing people suffering “intolerably” from an incurable physical position should have the option of assisted death.