  The $CP$ violation observables $S$ and $C$ in the decay channel $B^0
\!\rightarrow D^+ D^-$ are determined from a sample of proton-proton collisions
at center-of-mass energies of 7 and 8 TeV, collected by the LHCb experiment and
corresponding to an integrated luminosity of 3 fb$^{-1}$. The observable $S$
describes $CP$ violation in the interference between mixing and the decay
amplitude, and $C$ parametrizes direct $CP$ violation in the decay. The
following values are obtained from a flavor-tagged, decay-time-dependent
analysis: \begin{align*}
  S &= -0.54 \, ^{+0.17}_{-0.16} \, \text{(stat)} \pm 0.05 \,
\text{(syst)}\,,\newline
  C &= \phantom{-}0.26 \, ^{+0.18}_{-0.17} \, \text{(stat)} \pm 0.02 \,
\text{(syst)}\,. \end{align*} These values provide evidence for $CP$ violation
at a significance level of 4.0 standard deviations. The phase shift due to
higher-order Standard Model corrections is constrained to a small value of
\begin{align*}
  \Delta\phi = -0.16\,^{+0.19}_{-0.21}\,\text{rad}\,. \end{align*}
