Neural networks have recently become popular for a wide variety of uses, but have seen limited application in safety-critical domains such as robotics near and around humans. This is because it remains an open challenge to train a neural network to obey safety constraints. Most existing safety-related methods only seek to verify that already-trained networks obey constraints, requiring alternating training and verification. Instead, this work proposes a constrained method to simultaneously train and verify a feedforward neural network with rectified linear unit (ReLU) nonlinearities. Constraints are enforced by computing the network's output-space reachable set and ensuring that it does not intersect with unsafe sets; training is achieved by formulating a novel collision-check loss function between the reachable set and unsafe portions of the output space. The reachable and unsafe sets are represented by constrained zonotopes, a convex polytope representation that enables differentiable collision checking. The proposed method is demonstrated successfully on a network with one nonlinearity layer and approximately 50 parameters.