Upon the passage of any ordinance, the enrolled copy thereof, attested by the recorder, shall be submitted to the mayor by the recorder, and if the mayor approves the same the mayor shall write upon it “Approved,” with the date thereof and sign it with the name of office of the mayor. Thereupon, unless otherwise provided, such ordinance shall become a law and be of force and effect. If the mayor does not approve of the ordinance so submitted, the mayor must, within 10 days from the receipt thereof, return the same to the recorder with the reasons of the mayor for not approving it, and if the mayor does not so return it, such ordinance shall become a law as if the mayor had approved it. Upon the first meeting of the council after the return of an ordinance from the mayor not approved, the recorder shall deliver it to the council, with the message of the mayor, which must be read. The ordinance shall then be put upon its passage again, and if two-thirds of all the members constituting the council, as then provided by law, vote in the affirmative, it shall become a law without the approval of the mayor, and not otherwise.