In January, former communities secretary Eric Pickles refused planning permission for a two-phase hybrid application for the clearance and redevelopment of 5.97 hectares of the Victorian-built terraced houses in the Welsh Streets area, rejecting the advice of a planning inspector.
The plans would have seen the demolition of more than 400 homes and their replacement by 220 low density houses.
Speaking at the time, assistant mayor and cabinet member for housing, Ann O’Byrne, said that "the long-suffering residents of the Welsh Streets have now been waiting more than a decade for new homes and we are standing up against this unacceptable political interference from Whitehall and challenging the decision."
However, the council has now confirmed that it is no longer challenging the ruling.
A council spokesman said: "We still regard the secretary of state's decision as unreasonable and illogical.
"However, our priority rather than engaging in protracted legal arguments is to regenerate the Welsh Streets area and our objective all along has been to provide a sustainable and deliverable housing scheme.
"We will continue to consult with the residents on how best to achieve that."
Campaign group SAVE Britain’s Heritage has welcomed the move, protecting the area, which it says, surrounds the birthplace of Beatles drummer Ringo Starr.
The move, it said in a press release, will "hopefully pave the way for this area to be revived and the community restored."
SAVE has been campaigning for the Welsh Streets area to be restored and reoccupied for some years. Its director Clem Cecil said: "We are delighted that the appeal has been withdrawn.
"It would have been better for the houses, that have been standing empty all year, if this decision had been taken more expediently.
"However, we now look forward to working with all stakeholders and assisting in finding a working solution for the site."