Built in the United States to a modified version of the Charles F. Adams design.
All that steel is now off the Australian coast, unlike the exHMAS Melbourne that was sold for scrap to Korea and I suspect came back as Hyundai cars.
The US also sold Australia a fourth
Charles F. Adams-class destroyer in 1993, the ex-USS
Goldsborough, as a parts hulk. Aside from removing spare parts to keep the three
Perth-class ships running, some equipment was taken ashore as training materials so RAN trainees wouldn't have to fly to the US to train on it. While the ship was tied up the RAN painted the number 40 on the bow to fill the gap in the number sequence between
Hobart (D39) and
Brisbane (D41); after it had been stripped of anything deemed useful it was sent to the breakers in India.
The former HMAS
Melbourne was actually sold for scrap to the PRC in 1985; according to some reports her dismantling was not completed until 2002 as PLAN made a thorough study of the ship for their own nascent aircraft carrier program. It was probably more useful to them than the pair of half-arse ex-Soviet "heavy aviation cruisers" that they acquired a decade later and turned into tourist attractions. You're probably thinking of
Melbourne's sister ship HMAS
Sydney; she was sold to a South Korean scrapyard in 1975 after serving as a fast troop transport (the "Vung Tau Ferry") during Vietnam.
As a side note, while
Melbourne was built at the end of WWII and the RAN had been looking to replace her since the late 1950s, one of the reasons she was decommissioned in 1982 was the British Royal Navy had decided a year earlier that their new "through-deck cruiser" HMS
Invincible was surplus to requirements and offered her to the RAN cheap ($285 million AUD at the time). Then Argentina took over the Falklands and HMS
Invincible and HMS
Hermes were the only air cover the Royal Navy had, which caused them to withdraw the sale offer.