Single crystalline samples of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ with $x < 0.12$ have been grown and characterized via microscopic, thermodynamic and transport measurements. With increasing Co substitution, the thermodynamic and transport signatures of the structural (high temperature tetragonal to low temperature orthorhombic) and magnetic (high temperature non magnetic to low temperature antiferromagnetic) transitions are suppressed at a rate of roughly 15 K per percent Co. In addition, for $x ge 0.038$ superconductivity is stabilized, rising to a maximum $T_c$ of approximately 23 K for $x approx 0.07$ and decreasing for higher $x$ values. The $T - x$ phase diagram for Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ indicates that either superconductivity can exist in both low temperature crystallographic phases or that there is a structural phase separation. Anisotropic, superconducting, upper critical field data ($H_{c2}(T)$) show a significant and clear change in anisotropy between samples that have higher temperature structural phase transitions and those that do not. These data show that the superconductivity is sensitive to the suppression of the higher temperature phase transition.
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