Galaxies are not distributed randomly throughout space but are instead arranged in an intricate "cosmic web" of filaments and walls surrounding bubble-like voids. There is still no compelling observational evidence of a link between the structure of the cosmic web and how galaxies form within it. However, such a connection is expected on the basis of our understanding of the origin of galaxy angular momentum: disk galaxies should be highly inclined relative to the plane defined by the large-scale structure surrounding them. Using the two largest galaxy redshift surveys currently in existence (2dFGRS and SDSS) we show at the 99.7% confident level that these alignments do indeed exist: spiral galaxies located on the shells of the largest cosmic voids have rotation axes that lie preferentially on the void surface.