Jorge Luis Borges imagined the infinite Library of Babel in an eponymous short story. The Library fills space with hexagonal galleries, containing all possible books of a certain size. Borges' description of the connectivity between galleries is both incomplete and inconsistent. But, by relaxing two troublemaking constraints in a natural way, there is a way to connect galleries so that every hexagon contains a gallery, and every gallery is reachable by foot in a time proportional to the geometric distance from the starting gallery. A virtual 3-D version of such a Library is available on a Web site.