Using a sample of approximately 14,000 z > 2.1 quasars observed in the first year of the Baryon Oscillation Spectroscopic Survey (BOSS), we measure the three-dimensional correlation function of absorption in the Lyman-α forest. The angle-averaged correlation function of transmitted flux (F = e−τ) is securely detected out to comoving separations of 60 h−1Mpc, the first detection of flux correlations across widely separated sightlines. A quadrupole distortion of the redshift-space correlation function by peculiar velocities, the signature of the gravitational instability origin of structure in the Lyman-α forest, is also detected at high significance. We obtain a good fit to the data assuming linear theory redshift-space distortion and linear bias of the transmitted flux, relative to the matter fluctuations of a standard ΛCDM cosmological model (inflationary cold dark matter with a cosmological constant). At 95% confidence, we find a linear bias parameter 0.16 < b < 0.24 and redshift-distortion parameter 0.44 < β < 1.20, at central redshift z = 2.25, with a well constrained combination b(1+β) = 0.336±0.012. The errors on β are asymmetric, with β = 0 excluded at over 5σ confidence level. The value of β is somewhat low compared to theoretical predictions, and our tests on synthetic data suggest that it is depressed (relative to expectations for the Lyman-α forest alone) by the presence of high column density systems and metal line absorption. These results set the stage for cosmological parameter determinations from three-dimensional structure in the Lyman-α forest, including anticipated constraints on dark energy from baryon acoustic oscillations.