  The temperature-dependent optical reflectivity and complex transmissivity of
an epitaxially grown Ba(Fe$_{0.9}$Co$_{0.1}$)$_2$As$_2$ thin film were measured
and the optical conductivity and permittivity evaluated over a wide frequency
range. The opening of the superconducting gap $2\Delta_0 = 3.7$ meV below
$T_c\approx 20$ K is {\em directly} observed by a completely vanishing optical
conductivity. The temperature and frequency dependent electrodynamic properties
of Ba(Fe$_{0.9}$Co$_{0.1}$)$_2$As$_2$ in the superconducting state agree well
with the BCS predictions with no nodes in the order parameter. The spectral
weight of the condensate $1.94\times 10^7 {\rm cm}^{-2}$ corresponds to a
London penetration depth $\lambda_L=3600$ \AA.
