Deciphering paleoclimate on Mars has been a driving goal of Martian science for decades. Most research has addressed this issue by studying Mars' large polar layered deposits (PLDs) as a paleoclimate proxy, but the certainty to which we know the link between climate and orbit is debated. Here, we instead consider the record of other, smaller ice deposits located within craters separated from the PLDs using images from NASA's High Resolution Imaging Science Experiment camera and signal processing techniques. We show that the climate record in Burroughs Crater (72.3°S, 116.6°E) contains robust evidence of orbital forcing, with periodicities that have wavelengths of 15.6 and 6.5 m. The ratio of these dominant wavelengths is 2.4, the same as the ratio between the periods of Mars' obliquity changes and orbital precession. This result suggests orbital control of recent Mars climate, and would imply an average ice accumulation rate of 0.13 mm/yr over 4.5 Myr in this region.