We present a measurement of the spatial clustering of submillimetre galaxies (SMGs) at z = 1-3. Using data from the 870 micron LESS survey, we employ a novel technique to measure the cross-correlation between SMGs and galaxies, accounting for the full probability distributions for photometric redshifts of the galaxies. From the observed projected two-point cross-correlation function we derive the linear bias and characteristic dark matter (DM) halo masses for the SMGs. We detect clustering in the cross-correlation between SMGs and galaxies at the > 4 sigma level. For the SMG autocorrelation we obtain r_0 = 7.7 (+1.8,-2.3) h^-1 Mpc, and derive a corresponding DM halo mass of log(M_halo [h^-1 M_sun]) = 12.8 (+0.3,-0.5). Based on the evolution of DM haloes derived from simulations, we show that that the z = 0 descendants of SMGs are typically massive (~2-3 L*) elliptical galaxies residing in moderate- to high-mass groups (log(M_halo [h^-1 M_sun]) = 13.3 (+0.3,-0.5). From the observed clustering we estimate an SMG lifetime of ~100 Myr, consistent with lifetimes derived from gas consumption times and star-formation timescales, although with considerable uncertainties. The clustering of SMGs at z ~ 2 is consistent with measurements for optically-selected quasi-stellar objects (QSOs), supporting evolutionary scenarios linking starbursts and QSOs. Given that SMGs reside in haloes of characteristic mass ~ 6 x 10^12 h^-1 M_sun, we demonstrate that the redshift distribution of SMGs can be described remarkably well by the combination of two effects: the cosmological growth of structure and the evolution of the molecular gas fraction in galaxies. We conclude that the powerful starbursts in SMGs likely represent a short-lived but universal phase in massive galaxy evolution, associated with the transition between cold gas-rich, star-forming galaxies and passively evolving systems. [Abridged]
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