This study addresses discrete network design problems under traffic equilibrium conditions or DNDPs. Given a network and a budget, DNDPs aim to model all-or-nothing decisions such as link addition to minimize network congestion effects. Congestion is measured using traffic equilibrium theory where link travel times are modeled as convex flow-dependent functions and where users make selfish routing decisions. In this context, the collective route choice of users is a Wardropian equilibrium and DNDPs admit a bilevel optimization formulation where the leader represents the network designer, and the follower is a parameterized traffic assignment problem (TAP). This study introduces a novel and exact branch-and-price-and-cut (BPC) algorithm for DNDPs that exploits the structure of the problem and harnesses the potential of path-based formulations for column generation (CG). Leveraging the convexity and the separability of the objective function, we develop successive relaxations of the bilevel problem that lead to an efficient outer approximation scheme that relies on solving a sequence of linear programs. We combine this OA procedure with a CG approach whose pricing subproblem can be solved in polynomial-time. This scheme is embedded within a single tree BPC algorithm to determine lower bounds while upper bounds are computed by solving parameterized TAPs. Numerical experiments conducted on a range of DNDP instances based on three transportation networks reveal that our BPC algorithm significantly outperforms state-of-the-art methods for DNDPs. Notably, we close several open instances of the literature, and we show that our BPC algorithm can solve DNDP instances based on networks whose number of nodes and of commodities is one order of magnitude larger than any previously solved instance.