All contracts for the making of any public improvement, to be paid for in whole or in part by special assessment, and any work or public improvement, when the expense thereof shall exceed $1500, shall be let to the lowest responsible bidder in the manner hereinafter prescribed, such contract to be approved by the Mayor. But any such contract may be entered into by the proper officer without advertising for bids, and without such approval, by a vote of two-thirds of the City Council.
('72 Code, § 3.20(a))