We present the first measurement of the large-scale cross-correlation of Lyα forest absorption and Damped Lyman α systems (DLA), using the 9th Data Release of the Baryon Oscillation Spectroscopic Survey (BOSS). The cross-correlation is clearly detected on scales up to 40h-1Mpc and is well fitted by the linear theory prediction of the standard Cold Dark Matter model of structure formation with the expected redshift distortions, confirming its origin in the gravitational evolution of structure. The amplitude of the DLA-Lyα cross-correlation depends on only one free parameter, the bias factor of the DLA systems, once the Lyα forest bias factors are known from independent Lyα forest correlation measurements. We measure the DLA bias factor to be bD = (2.17±0.20)βF0.22, where the Lyα forest redshift distortion parameter βF is expected to be above unity. This bias factor implies a typical host halo mass for DLAs that is much larger than expected in present DLA models, and is reproduced if the DLA cross section scales with halo mass as Mhα, with α = 1.1±0.1 for βF = 1. Matching the observed DLA bias factor and rate of incidence requires that atomic gas remains extended in massive halos over larger areas than predicted in present simulations of galaxy formation, with typical DLA proper sizes larger than 20 kpc in host halos of masses ~ 1012Msun. We infer that typical galaxies at z simeq 2 to 3 are surrounded by systems of atomic clouds that are much more extended than the luminous parts of galaxies and contain ~ 10% of the baryons in the host halo.