At any time after the funds for the proposed improvement, or any part thereof, are in the hands of the city treasurer, the city council may, without calling for bids, order by resolution, adopted by a vote of two-thirds of all its members, that the city itself execute the work of such improvement, or any part thereof, in accordance with the plans and specifications adopted for such work or for such part thereof, and employ the labor, and provide the materials, appliances, supplies or illuminating agent necessary therefor. In such case the costs and expenses of such work, or of such part thereof, shall be paid out of the aforesaid funds, except that if such costs and expenses exceed the amount of the respective estimates provided for in Section 18041 of the Streets and Highways Code, then such excess shall be paid out of the moneys in the general fund in the city treasury.