We present photometry and spectroscopy of the peculiar Type II supernova SN 2010jp, also named PTF10aaxi. The light curve exhibits a linear decline with a relatively low peak absolute magnitude of only −15.9 (unfiltered), and a low radioactive decay luminosity at late times that suggests a low synthesized nickel mass of about 0.003 M⊙ or less. Spectra of SN 2010jp display an unprecedented triple-peaked Hα line profile, showing: (1) a narrow central component that suggests shock interaction with a dense circumstellar medium (CSM); (2) high-velocity blue and red emission features centered at −12,600 and +15,400 km s−1; and (3) very broad wings extending from −22,000 to +25,000 km s−1. We propose that this line profile indicates a bipolar jet-driven explosion, with the central component produced by normal SN ejecta and CSM interaction at mid and low latitudes, while the high-velocity bumps and broad line wings arise in a nonrelativistic bipolar jet. Jet-driven SNe II are predicted for collapsars resulting from a wide range of initial masses above 25 M⊙, especially at the sub-solar metallicity consistent with the SN host environment. It also seems consistent with the apparently low 56Ni mass that may accompany black hole formation. We speculate that the jet survives to produce observable signatures because the star's H envelope was very low mass, having been mostly stripped away by the previous eruptive mass loss.