We have performed a survey for substellar companions to young brown dwarfs in the Taurus star-forming region using the Wide Field Planetary Camera 2 on board the Hubble Space Telescope. In these data, we have discovered a candidate companion at a projected separation of 0105 from one of the brown dwarfs, corresponding to 15 AU at the distance of Taurus. To determine if this object is a companion, we have obtained images of the pair at a second epoch with the adaptive optics system at Gemini Observatory. The astrometry from the Hubble and Gemini data indicates that the two objects share similar proper motions and thus are likely companions. We estimate a mass of 5-10 M _Jup for the secondary based on a comparison of its bolometric luminosity to the predictions of theoretical evolutionary models. This object demonstrates that planetary-mass companions to brown dwarfs can form on a timescale of τ 1 Myr. Companion formation on such a rapid timescale is more likely to occur via gravitational instability in a disk or fragmentation of a cloud core than through core accretion. The Gemini images also reveal a possible substellar companion (ρ = 023) to a young low-mass star that is 124 from the brown dwarf targeted by Hubble. If these four objects comprise a quadruple system, then its hierarchical configuration would suggest that the fragmentation of molecular cloud cores can produce companions below 10 M _Jup.