We investigate the temperature dependence of the lower critical field H c1 (T ) of a high-quality FeSe single crystal under static magnetic fields H parallel to the c axis. The temperature dependence of the first vortex penetration field has been experimentally obtained by two independent methods and the corresponding H c1 (T ) was deduced by taking into account demagnetization factors. A pronounced change in the H c1 (T) curvature is observed, which is attributed to anisotopic s-wave or multiband superconductivity. The London penetration depth λ ab (T ) calculated from the lower critical field does not follow an exponential behavior at low temperatures, as it would be expected for a fully gapped clean s-wave superconductor. Using either a two-band model with s-wave-like gaps of magnitudes 1 = 0.41 ± 0.1 meV and 2 = 3.33 ± 0.25 meV or a single anisotropic s-wave order parameter, the temperature dependence of the lower critical field H c1 (T ) can be well described. These observations clearly show that the superconducting energy gap in FeSe is nodeless.