  The temperature dependence of the real part of the microwave complex
conductivity at 17.9 GHz obtained from surface impedance measurements of two
c-axis oriented MgB2 thin films reveals a pronounced maximum at a temperature
around 0.6 times the critical temperature. Calculations in the frame of a
two-band model based on Bardeen-Cooper-Schrieffer (BCS) theory suggest that
this maximum corresponds to an anomalous coherence peak resembling the two-gap
nature of MgB2. Our model assumes there is no interband impurity scattering and
a weak interband pairing interaction, as suggested by bandstructure
calculations. In addition, the observation of a coherence peak indicates that
the pi-band is in the dirty limit and dominates the total conductivity of our
films
