  The effects of Co-substitution on Ba(Ni$_{1-x}$Co$_x$)$_2$As$_2$ ($0\leq
x\leq 0.251$) single crystals grown out of Pb flux are investigated via
transport, magnetic, and thermodynamic measurements. BaNi$_2$As$_2$ exhibits a
first order tetragonal to triclinic structural phase transition at $T_s=137 K$
upon cooling, and enters a superconducting phase below $T_c=0.7 K$. The
structural phase transition is sensitive to cobalt content and is suppressed
completely by $x\geq0.133$. The superconducting critical temperature, $T_c$,
increases continuously with $x$, reaching a maximum of $T_c=2.3 K$ at the
structural critical point $x=0.083$ and then decreases monotonically until
superconductivity is no longer observable well into the tetragonal phase. In
contrast to similar BaNi$_2$As$_2$ substitutional studies, which show an abrupt
change in $T_c$ at the triclinic-tetragonal boundary that extends far into the
tetragonal phase, Ba(Ni$_{1-x}$Co$_x$)$_2$As$_2$ exhibits a dome-like phase
diagram centered around the first-order critical point. Together with an
anomalously large heat capacity jump $\Delta C_e/\gamma T\sim 2.2$ at optimal
doping, the smooth evolution of $T_c$ in the Ba(Ni$_{1-x}$Co$_x$)$_2$As$_2$
system suggests a mechanism for pairing enhancement other than phonon
softening.
