An experimental technique for searching for the 2K capture in 124Xe with a large low-background copper proportional counter is described. Such an experiment is conducted at the Baksan Neutrino Observatory of the Institute for Nuclear Research of the Russian Academy of Sciences. The experimental setup is located in the Low-Background Deep-Level Laboratory at a depth of 4900 m.w.e., where the flux of muons of cosmic rays is suppressed by a factor of 107 relative to that at the Earth’s surface. The setup incorporates a proportional counter and low-background shielding (18 cm of copper, 15 cm of lead, and 8 cm of borated polyethylene). The results of processing the data obtained in 5 months of live measurement time are presented. A new limit on the half-life of 124Xe with respect to the 2K capture is set at the level of 2.5 × 1021 years.