Large-scale correlations in the orientations of galaxies can result from alignments in their angular momentum vectors. These alignments arise from the tidal torques exerted on neighboring protogalaxies by the smoothly varying shear field. We compute the predicted amplitude of such ellipticity correlations using the Zeldovich approximation for a realistic distribution of galaxy shapes. Weak gravitational lensing can also induce ellipticity correlations, since the images of neighboring galaxies will be distorted coherently. On comparing these two effects that induce shape correlations, we find that for current weak-lensing surveys with a median redshift of zm=1, the intrinsic signal is of the order of 1%-10% of the measured signal. However, for shallower surveys with zm<=0.3, the intrinsic correlations dominate over the lensing signal. The distortions induced by lensing are curl-free, whereas those resulting from intrinsic alignments are not. This difference can be used to disentangle these two sources of ellipticity correlations.