Background: Time-reversal-invariance violation, or equivalently $CP$ violation, may explain the observed cosmological baryon asymmetry as well as indicate physics beyond the Standard Model. In the decay of polarized neutrons, the triple correlation $D\ensuremath{\langle}{\stackrel{P\vec}{J}}_{n}\ensuremath{\rangle}/{J}_{n}\ifmmode\cdot\else\textperiodcentered\fi{}({\stackrel{P\vec}{\ensuremath{\beta}}}_{e}\ifmmode\times\else\texttimes\fi{}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{\ensuremath{\nu}})$ is a parity-even, time-reversal-odd observable that is uniquely sensitive to the relative phase of the axial-vector amplitude with respect to the vector amplitude. The triple correlation is also sensitive to possible contributions from scalar and tensor amplitudes. Final-state effects contribute to $D$ at the level of 10${}^{\ensuremath{-}5}$ and can be calculated with a precision of 1$%$ or better.Purpose: We have improved the sensitivity to $T$-odd, $P$-even interactions in nuclear $\ensuremath{\beta}$ decay.Methods: We measured proton-electron coincidences from decays of longitudinally polarized neutrons with a highly symmetric detector array designed to cancel the time-reversal-even, parity-odd Standard-Model contributions to polarized neutron decay. Over 300 million proton-electron coincidence events were used to extract $D$ and study systematic effects in a blind analysis.Results: We find $D=[\ensuremath{-}0.94\ifmmode\pm\else\textpm\fi{}1.89(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.97(\mathrm{sys})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. This differs from the result of our recent paper [Phys. Rev. Lett. 107, 102301 (2011)] due to refinement of corrections for background and backscattering.Conclusions: This is the most sensitive measurement of $D$ in nuclear $\ensuremath{\beta}$ decay. Our result can be interpreted as a measurement of the phase of the ratio of the axial-vector and vector coupling constants (${C}_{A}/{C}_{V}=|\ensuremath{\lambda}|{e}^{i{\ensuremath{\varphi}}_{AV}}$) with ${\ensuremath{\varphi}}_{AV}=180.{012}^{\ensuremath{\circ}}\ifmmode\pm\else\textpm\fi{}0.{028}^{\ensuremath{\circ}}$ (68$%$ confidence level). This result can also be used to constrain time-reversal-violating scalar and tensor interactions that arise in certain extensions to the Standard Model such as leptoquarks.