The library board shall be composed of nine members appointed by the mayor and approved by the council, no more than three of whom shall reside outside the city. The library board shall have one member of the council on the board. The term of office of the members of the library board shall be for three years, but it shall be so arranged that only two members shall complete their full term of office in any one year, thereby maintaining an experienced member on the board at all times. This provision shall apply only to those persons appointed to the board other than the councilmembers so appointed. The councilmember appointed to the board shall be appointed for one year, and only newly elected councilmembers shall be appointed to the board so that there may be no candidate serving on the board during the last year of his term.