Any city or town with a population of three hundred thousand or less at the time of annexation may become a part of any rural county library district, island library district, or intercounty rural library district lying contiguous thereto by annexation in the following manner: The inclusion of such a city or town may be initiated by the adoption of an ordinance by the legislative authority thereof stating its intent to join the library district and finding that the public interest will be served thereby. Before adoption, the ordinance shall be submitted to the library board of the city or town for its review and recommendations. If no library board exists in the city or town, the state librarian shall be notified of the proposed ordinance. If the board of trustees of the library district concurs in the annexation, notification thereof shall be transmitted to the legislative authority or authorities of the counties in which the city or town is situated.