We report the discovery of a resolved (0″.9) substellar companion to a member of the 1-5 Myr Taurus star-forming region. The host star (2M0437) is a single mid-M type (Teff≈3100 K) dwarf with a position, space motion, and color-magnitude that support Taurus membership, and possible affiliation with a ∼2.5 Myr-old sub-group. A comparison with stellar models suggests a 2-5 Myr age and a mass of 0.15-0.18M⊙. Although K2 detected quasi-periodic dimming from close-in circumstellar dust, the star lacks detectable excess infrared emission from a circumstellar disk and its Hα emission is not commensurate with accretion. Astrometry based on three years of AO imaging shows that the companion (2M0437b) is co-moving, while photometry of two other sources at larger separation indicates they are likely heavily-reddened background stars. A comparison of the luminosity of 2M0437b with models suggests a mass of 3-5 MJUP , well below the deuterium burning limit, and an effective temperature of 1400-1500 K, characteristic of a late L spectral type. The H-K color is redder than the typical L dwarf, but comparable to other directly detected young planets, e.g. those around HR 8799. The discovery of a super-Jupiter around a very young, very low mass star challenges models of planet formation by either core accretion (which requires time) or disk instability (which requires mass). We also detected a second, co-moving, widely-separated (75″) object which appears to be a heavily-extincted star. This is certainly a fellow member of this Taurus sub-group and statistically likely to be a bound companion.