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 $^56$Ni 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.