Hubble Space Telescope (HST) V- and I-band observations show that the gravitational lens B1359+154 consists of six images of a single zs=3.235 radio source and its star-forming host galaxy, produced by a compact group of galaxies at zl~=1. Very Long Baseline Array (VLBA) observations at 1.7 GHz strongly support this conclusion, showing six compact cores with similar low-frequency radio spectra. B1359+154 is the first example of galaxy-scale gravitational lensing in which more than four images are observed of the same background source. The configuration is due to the unique lensing mass distribution: three primary lens galaxies lying on the vertices of a triangle separated by 0.7"~=4 h-1 kpc, inside the 1.7" diameter Einstein ring defined by the radio images. The gravitational potential has additional extrema within this triangle, creating a pair of central images that supplement the ``standard'' four-image geometry of the outer components. Simple mass models, consisting of three lens galaxies constrained by HST and VLBA astrometry, naturally reproduce the observed image positions but must be finely tuned to fit the flux densities. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.