The influence of physisorbed krypton on the lifetime of positrons trapped at copper surfaces has been investigated, using bulk Cu samples containing Kr-filled micrometre-size cavities. A decrease in the trapped-positron lifetime with decreasing temperature is attributed to adsorption of Kr on the cavity surfaces, in accordance with recent theoretical results of Jensen and Nieminen (1987). The results demonstrate that positrons are trapped at the cavity surfaces. In addition, the trapping of positrons into the cavities is shown to be limited both by their rate of diffusion in the bulk, and their rate of transition into the cavities. The transition rate and its temperature dependence are estimated.