We report on a survey for strong (rest equivalent width W$_r$>=1 ̊A), intervening Mg II systems along the sight lines to long-duration gamma-ray bursts (GRBs). The GRB spectra that comprise the survey have a heterogeneous mix of resolution and wavelength coverage, but we implement a strict, uniform set of search criteria to derive a well-defined statistical sample. We identify 14 strong Mg II absorbers along 14 GRB sight lines (nearly every sight line exhibits at least one absorber) with spectra covering a total path length Δz=15.5 at a mean redshift z̄=1.1. In contrast, the predicted incidence of such absorber systems along the same path length to quasar sight lines is only 3.8. The roughly 4 times higher incidence along GRB sight lines is inconsistent with a statistical fluctuation at greater than 99.9% c.l. Several effects could explain the result: (1) dust within the Mg II absorbers obscures faint quasars giving a lower observed incidence along quasar sight lines, (2) the gas is intrinsic to the GRB event, and (3) the GRBs are gravitationally lensed by these absorbers. We present strong arguments against the first two effects and also consider lensing to be an unlikely explanation. The results suggest that at least one of our fundamental assumptions underpinning extragalactic absorption line research is flawed.