  We consider the far-infrared and the microwave conductivities of a two-band
superconductor with non-magnetic impurities. The strong coupling expressions
for the frequency and temperature dependent conductivity of a two-band
superconductor are developed assuming isotropic bands and interactions. Our
numerical results obtained using realistic interaction parameters for MgB$_{2}$
are compared with experiments on this compound. We find that the available
experimental results for the far-infrared conductivity of MgB$_{2}$ are
consistent with multi-band superconductivity in the presence of a sufficiently
strong interband impurity scattering. On the other hand, our numerical results
for the microwave conductivity in the superconducting state indicate that the
experimental results obtained on samples with the highest transition
temperature $T_{c}$ are consistent with a low interband impurity scattering
rate but depend sensitively on the ratio of the total scattering rates in the
two bands. For the $\pi$-band scattering rate $\gamma_{\pi}$ not greater than
the $\sigma$-band scattering rate $\gamma_{\sigma}$ there is a single, broad,
low-temperature (at about 0.5$T_{c}$) coherence peak in the microwave
conductivity. For $\gamma_{\pi}/\gamma_{\sigma}$=4--7 a high-temperature (at
about 0.9$T_{c}$) coherence peak is dominant, but there is also a
low-temperature peak/shoulder resulting from the contribution of the $\pi$-band
carriers to the microwave conductivity. For $\gamma_{\pi}/\gamma_{\sigma}\gg$1
only the high-temperature coherence peak should be observable.
