Conn. Gen. Stat. § 2-5: Holding of office by members of the General Assembly.

No member of the General Assembly shall, during the term for which he is elected, be nominated or appointed by the Governor, the General Assembly or any other appointing authority of this state or serve or be elected to any position in the Judicial, Legislative or Executive Department of the state government including any commission established by any special or public act of the General Assembly, except that the provisions of this section shall not apply where it is expressly provided by law that a member of the General Assembly as such shall be nominated or appointed to any board, commission, council or other agency in the legislative department, and except that the provisions of this section shall not apply to a member-elect.