The bill, which passed the Upper House earlier this month, was ushered through the House of Assembly on its second reading vote 33-5.
A third and final vote on the legislation is scheduled for June 9 where lawmakers will debate amendments to the bill.
A change in the make-up of the Lower House means the legislation looks poised to become law.
Liberal MPs Vincent Tarzia and Adrian Pederick, Labor MPs Tom Koutsantonis and Michael Brown and Independent MP Sam Duluk were the only votes against in the second reading of the bill.
Labor MLC Kyam Maher introduced the legislation to the Upper House in December last year, with the bill representing the 17th attempt in 25 years to legalise euthanasia in South Australia.
The bill is modelled on laws passed in Victoria and includes 68 safeguards and a provision that someone applying to end their life early due to terminal illness must have been a resident in SA for at least 12 months.
The second reading vote followed rallies on the step of Parliament House by both proponents and opponents of the bill attended by hundreds of people.
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