An experiment has been performed in the Kolar Gold Fields in Southern India to search for the interaction products of cosmic ray neutrinos. In the course of four years operation of the detectors at a depth of 2316 m of rock, some 165 particles were recorded which were attributed to muons of atmospheric (as distinct from neutrino-) origin and the present paper describes the results of measurements on these particles. The measured vertical intensity at the depth in question (2316 m of Kolar rock corresponding to 7.6 x 105 g cm-2 of standard rock) is (1.1 ± 0.2) 10-6 m-2 s-1 sr-1. Including measurements at shallower depths by other workers, from sites elsewhere in the same Gold Fields, a best fit to the data gives the relation I(h) = 7.73 x 10-3 exp (–h/790) m-2 sr-1 s-1 for depth range 4000 < h < 9500 hg cm-2 Kolar rock (1 hg cm-2 = 102 g cm-2). An approximate estimate has been made of the mean energy of the atmospheric muons at the depth of operation; its value, ca. 330 GeV, is not inconsistent with the expected value.