We report on measurements of the temperature dependence of the magnetic penetration depth down to 0.04 K in a high-quality sample of the beta-pyrochlore KOs2O6 (Tc=9.65 K) with a spin-frustrated lattice. We observe temperature-independent behavior below T approximately 0.3Tc, which is firm evidence for the presence of an isotropic superconducting gap in this material. In the whole temperature range the superfluid density is very well described, without the need of adjustable parameters, by a strong-coupling extension of the BCS model for an isotropic gap. Thus, the penetration depth results indicate that KOs2O6 is a strong-coupling superconductor with a fully developed energy gap. No effect of the second phase transition taking place at Tp=7.5 K was observed on the penetration depth, which suggests that the Cooper pairs remain unperturbed across this transition.