From the library's website:
The library received taxpayer support in 1898, independence from the local school board in 1913, and a permanent location in 1916.
Wealthy industrialist Andrew Carnegie donated millions of dollars worldwide establishing public libraries, and New Castle was among 64 Indiana communities receiving a “Carnegie” library. This library building opened in 1916 with 4,000 volumes and 1,803 registered borrowers. 
From The Library Journal of March 1916:
Indiana. Newcastle. Newcastle's new Public Library was opened and dedicated Jan. 17. Demarchus C. Brown, state librarian, gave the principal address. The library was made possible by the untiring efforts of Miss Lois Compton, who started the movement which resulted in an appropriation of $20,000 from the Carnegie Corporation.