The third-order elastic constants of calcite at 0\ifmmode^\circ\else\textdegree\fi{}C have been determined by measuring the stress and temperature dependence of sound velocities in it by means of an improved pulse-superposition method (average sensitivity of 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$). Within the small temperature range considered (about 2\ifmmode^\circ\else\textdegree\fi{}C), a nonlinear temperature dependence which varies with pressure has been clearly observed. Out of 14 independent third-order moduli, ${C}_{114}$ and ${C}_{134}$ are definitely positive, and all the others are negative, with ambiguities for ${C}_{124}$ and ${C}_{444}$. The approximate magnitude is equally large for ${C}_{111}$, ${C}_{222}$, and ${C}_{333}$, intermediate for ${C}_{112}$, ${C}_{113}$, ${C}_{114}$, ${C}_{133}$, ${C}_{155}$, ${C}_{344}$, and very small for ${C}_{123}$, ${C}_{124}$, ${C}_{134}$, ${C}_{144}$, ${C}_{444}$. The pressure derivative of the bulk modulus calculated using these constants is in reasonable agreement with Bridgman's data for the change in compressibility with pressure. The contribution of the ion-core, short-range repulsive interaction ${{C}_{\mathrm{ijk}}}^{R}$ to the third-order elastic constants has been evaluated for the carbonate and nitrate crystals of the calcite type using an inverse-power potential. The repulsive contributions were found to be predominant over the other contributions to almost all the third-order constants. Under the assumption that the remaining contribution is the electrostatic interaction alone, and using the experimental data for calcite, complete sets of the third-order constants have been estimated for other carbonate crystals.