N.Y. ACG Law § 205

§ 205. Vacancies in elective offices. Vacancy in the offices of\nsheriff, coroner, county clerk, register, district attorney, and county\ntreasurer shall be filled by the governor with the advice and consent of\nthe senate if in session. When a vacancy shall occur, otherwise than by\nexpiration of term, in the office of judge of the county court,\nsurrogate, or judge of the family court outside the city of New York, it\nshall be filled for a full term at the next general election held not\nless than three months after such vacancy occurs and, until the vacancy\nshall be so filled, the governor by and with the consent of the senate,\nif the senate shall be in session, or, if the senate not be in session\nthe governor may fill such vacancy by an appointment which shall\ncontinue until and including the last day of December next after the\nelection at which the vacancy shall be filled. All other vacancies in an\nelective county office shall be filled by the board of supervisors.\n