We report the discovery of the near-infrared and optical afterglow of the short-duration gamma-ray burst GRB 070724A . The afterglow is detected in iJHK _ { s } observations starting 2.3 hr after the burst with K _ { s } = 19.59 \pm 0.16 mag and i = 23.79 \pm 0.07 mag , but is absent in images obtained 1.3 years later . Fading is also detected in the K _ { s } -band between 2.8 and 3.7 hr at a 4 \sigma significance level . The optical/near-IR spectral index , \beta _ { O,NIR } \approx - 2 , is much redder than expected in the standard afterglow model , pointing to either significant dust extinction , A _ { V } ^ { host } \approx 2 mag , or a non-afterglow origin for the near-IR emission . The case for extinction is supported by a shallow optical to X-ray spectral index , consistent with the definition for ‘ ‘ dark bursts ’ ’ , and a normal near-IR to X-ray spectral index . Moreover , a comparison to the optical discovery magnitudes of all short GRBs with optical afterglows indicates that the near-IR counterpart of GRB 070724A is one of the brightest to date , while its observed optical emission is one of the faintest . In the context of a non-afterglow origin , the near-IR emission may be dominated by a mini-supernova , leading to an estimated ejected mass of M \sim 10 ^ { -4 } M _ { \odot } and a radioactive energy release efficiency of f \sim 5 \times 10 ^ { -3 } ( for v \sim 0.3 c ) . However , the mini-SN model predicts a spectral peak in the UV rather than near-IR , suggesting that this is either not the correct interpretation or that the mini-SN models need to be revised . Finally , the afterglow coincides with a star forming galaxy at z = 0.457 , previously identified as the host based on its coincidence with the X-ray afterglow position ( \sim 2 ^ { \prime \prime } radius ) . Our discovery of the optical/near-IR afterglow makes this association secure , and furthermore localizes the burst to the outskirts of the galaxy , with an offset of 4.8 \pm 0.1 kpc relative to the host center . At such a large offset , the possible large extinction points to a dusty environment local to the burst and rules out a halo or intergalactic origin .