We report the observation of a gravitational-wave signal produced by the coalescence of two stellar-mass black holes. The signal, GW151226, was observed by the twin detectors of the Laser Interferometer Gravitational-Wave Observatory (LIGO) on December 26, 2015 at 03:38:53 UTC. The signal was initially identified within 70 s by an online matched-filter search targeting binary coalescences. Subsequent off-line analyses recovered GW151226 with a network signal-to-noise ratio of 13 and a significance greater than 5σ. The signal persisted in the LIGO frequency band for approximately 1 s, increasing in frequency and amplitude over about 55 cycles from 35 to 450 Hz, and reached a peak gravitational strain of 3.4-0.9+0.7×10-22. The inferred source-frame initial black hole masses are 14.2-3.7+8.3M and 7.5-2.3+2.3M, and the final black hole mass is 20.8-1.7+6.1M. We find that at least one of the component black holes has spin greater than 0.2. This source is located at a luminosity distance of 440-190+180 Mpc corresponding to a redshift of 0.09-0.04+0.03. All uncertainties define a 90% credible interval. This second gravitational-wave observation provides improved constraints on stellar populations and on deviations from general relativity.