This paper presents a mathematical programming model and solution method for the path-constrained traffic assignment problem, in which route choices simultaneously follow the Wardropian equilibrium principle and yield the distance constraint imposed on the path. This problem is motivated by the need for modeling distance-restrained electric vehicles in congested networks, but the resulting model and solution method can be applied to various conditions with similar path-based constraints. The equilibrium conditions of the problem reveal that any path cost in the network is the sum of corresponding link costs and a path-specific out-of-range penalty term. The suggested method, based on the classic Frank-Wolfe algorithm, incorporates an efficient constrained shortest-path algorithm as its subroutine. This algorithm fully exploits the underlying network structure of the problem and is relatively easy to implement. Numerical results from the examples of problems provided show how the equilibrium conditions are reshaped by the path constraint and how the traffic flow patterns are affected by different constraint tightness levels.