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 $2Delta_0 = 3.7$ meV below $T_capprox 20$ 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 $1.94times 10^7 {rm cm}^{-2}$ corresponds to a London penetration depth $lambda_L=3600$ AA.
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