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 $5\sigma$. 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 $3.4_{-0.9}^{+0.7}\times {10}^{-22}$. The inferred source-frame initial black hole masses are ${14.2}_{-3.7}^{+8.3}{M}_{\odot }$ and $7.5_{-2.3}^{+2.3}{M}_{\odot }$, and the final black hole mass is ${20.8}_{-1.7}^{+6.1}{M}_{\odot }$. 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 $440_{-190}^{+180}\mathrm{Mpc}$ 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.