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 \(440_{-190}^{+180}\) Mpc corresponding to a redshift \(0.09_{-0.04}^{+0.03}\). All uncertainties define a 90 % credible interval. This second gravitational-wave observation provides improved constraints on stellar populations and on deviations from general relativity.