That is not an easy question to answer. Several libraries might be considered for the title; however, the Darby Free Library in Darby, Pennsylvania is probably the winner.
* Added - the Public Library solely supported by public contributions, a free public library, was opened in Dublin, New Hampshire in 1822. It lays claim to being the first in the US.
Benjamin Franklin
sidney public library
Scoville Public Library in Salisbury, Connecticut claims to be the oldest public library; they have a plaque.
The Library Company is America's first successful lending library and oldest cultural institution. It was founded in 1731 by Benjamin Franklin as a subscription library supported by its shareholders, as it is to this day. See: http://www.librarycompany.org/about/index.htm -- Libarian, University of Hawaii at Manoa
In 1890, the first Illinois public library opened in Princeton, Illinois. The library began as a dream of Nehemiah Matson in 1885.
The first public libary in England was in the year 1608
The address of the Quemado Public Library is: 19791 Us 277, Quemado, 78877 M
The first public library that was opened in 1747 was located in the town of Saulieu. Saulieu is in the eastern part of France.
The address of the Bartlett Public Library is: 1313 Us Rte 302, Bartlett, 03812 0399
Illinois first public libary opened in this town
Plymouth
The first free public library supported by taxation was the Peterborough, New Hampshire Public Library, established on April 9, 1833.