We present the first measurement of the large-scale cross-correlation of Ly \alpha  forest absorption and Damped Lyman \alpha systems ( DLA ) , using the 9th Data Release of the Baryon Oscillation Spectroscopic Survey ( BOSS ) . The cross-correlation is clearly detected on scales up to 40 h ^ { -1 } { Mpc } 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-Ly \alpha  cross-correlation depends on only one free parameter , the bias factor of the DLA systems , once the Ly \alpha  forest bias factors are known from independent Ly \alpha  forest correlation measurements . We measure the DLA bias factor to be b _ { D } = ( 2.17 \pm 0.20 ) \beta _ { F } ^ { 0.22 } , where the Ly \alpha  forest redshift distortion parameter \beta _ { F } 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 M _ { h } ^ { \alpha } , with \alpha = 1.1 \pm 0.1 for \beta _ { F } = 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 \sim 10 ^ { 12 } M _ { \odot } . 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 \sim 10 \% of the baryons in the host halo .