  We have measured the temperature dependence of the microwave surface
impedance Z_s = R_s + i\Omega\mu_0 of a MgB2 film at a frequency \Omega /2\Pi
of 18 GHz employing a dielectric resonator technique. We found that the
temperature dependence of the magnetic field penetration depth \Lamda can be
fitted by \Lamda(T)= \Lamda(0) [1-(T/Tc)^2] ^(-1/2) with \Lamda(0)=(260 +/-
20)nm. The absolute value of \Lamda(0) was confirmed by direct measurements
employing submillimeter wave transmission spectroscopy at 430 GHz. The analysis
of the \Lamda(T) data below Tc/2 revealed significant deviations from the
quadratic temperature dependence. In contrast, we found that an exponential
temperature dependence fits the experimental data within the statistical
measurement error for temperature changes of \Lamda of +/- 0.4 nm. This
observation indicates thermal excitation of quasiparticles over a finite energy
gap of (3.3 +/- 0.3) meV corresponding to \Delta /kTc = 1.2 +/- 0.1. Our
results strongly supports a multipgap or a strongly anisotropic gap and the
absence of nodes in the gap function.
