First-order exact solutions of the nonlinear Schrödinger equation in the normal-dispersion regime

``First-order'' exact solutions of the nonlinear Schr\"odinger equation (NLSE) with positive group-velocity dispersion are obtained. We find a three-parameter family of solutions that are finite everywhere; particular cases include periodic solutions expressed in terms of elliptic Jacobi functions, stationary periodic solutions, and solutions describing the collision or excitation of two dark solitons with equal amplitudes. A classification of solutions using the plane of their parameters, a geometrical description on the complex plane, and physical interpretations of the solutions obtained are given. A simple relation, which permits transformation of the solutions of the NLSE in the anomalous-dispersion regime into solutions of the NLSE in the normal-dispersion regime, is also discussed.