We have measured direct photons for $p_T<5~$GeV/$c$ in minimum bias and
0\%--40\% most central events at midrapidity for Cu$+$Cu collisions at
$\sqrt{s_{_{NN } } }=200$ GeV. The $e^{+}e^{-}$ contribution from quasi-real
direct virtual photons has been determined as an excess over the known hadronic
contributions in the $e^{+}e^{-}$ mass distribution. A clear enhancement of
photons over the binary scaled $p$$+$$p$ fit is observed for $p_T<4$ GeV/$c$ in
Cu$+$Cu data. The $p_T$ spectra are consistent with the Au$+$Au data covering a
similar number of participants. The inverse slopes of the exponential fits to
the excess after subtraction of the $p$$+$$p$ baseline are
285$\pm$53(stat)$\pm$57(syst)~MeV/$c$ and 333$\pm$72(stat)$\pm$45(syst)~MeV/$c$
for minimum bias and 0\%--40\% most central events, respectively. The rapidity
density, $dN/dy$, of photons demonstrates the same power law as a function of
$dN_{\rm ch}/d\eta$ observed in Au$+$Au at the same collision energy.