Wellington City Council estimates the cost of temporarily quake-proofing a multi-storied external lift shaft could run into five figures.
The 10-storey structure is a remnant of an unfinished building project from 20 years ago.
Work to brace it to the side of a parking building began on Friday after it was decided it could fall in an aftershock from last Sunday's magnitude 6.5 earthquake.
The work is expected to be finished by Monday.
The council says it will bear the cost initially, but will want its money back from the owner or owners - once it has tracked down who they are.
People in four buildings in neighbouring buildings were advised to leave on Friday. Lukes Lane and Opera House Lane were closed.