We report an investigation of the fundamental absorption edge of InSe under high-resolution conditions. We resolve three components of the direct exciton series and obtain an effective Rydberg energy of 14.5 meV. From this value an effective mass ($m=0.10 {m}_{0}$) of electrons in the $\ensuremath{\Gamma}$ minimum of the conduction band is obtained. We analyze the absorption coefficient with a three-dimensional model and find a remarkable agreement. We deduce an interband matrix element in polarization $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\perp}\stackrel{\ensuremath{\rightarrow}}{\mathrm{C}}:{P}_{\ensuremath{\perp}}^{2}=0.6$ eV. Next we investigate the temperature dependence of the fundamental absorption edge. We find a strong interaction with a 14-meV phonon which accounts for (i) the shift of the band-gap energy in the full temperature range between liquid-helium temperature and 300\ifmmode^\circ\else\textdegree\fi{}K and (ii) the temperature dependence of the broadening parameter (exciton lifetime). A simple analytical expression is obtained which accounts for the temperature dependence of the band gap and the $n=1$ exciton structure. Last, we deduce the electron-phonon coupling constant.