Nonlinear dynamic system identification using artificial neural networks has garnered attention due to its manifold potential applications in digital twins of engineering systems. However, purely data-driven machine learning algorithms typically require large amounts of training data and lack robustness, as they are prone to making physically implausible forecasts and can exhibit spurious instabilities. Physics-guided machine learning seeks to resolve these challenges by integrating physical biases into the model architectures. This approach promises enhanced accuracy, robustness, and interpretability while reducing data requirements. The present work presents stable port-Hamiltonian Neural Networks (sPHNNs) - a physics-guided machine learning architecture for identifying nonlinear dynamic systems. The model leverages the port-Hamiltonian framework, enabling the representation of both conservative and dissipative dynamic systems with external inputs. By learning a constrained Hamiltonian energy function from data, the approach incorporates physical biases and stability guarantees. A balance equation regarding the learned energy is satisfied by design. This balance states that the stored energy within a system can, at most, grow by the amount of energy provided via the inputs. Stability is ensured by constraining the Hamiltonian to be a convex, positive definite Lyapunov function. In the energy-conserving case, this approach guarantees the existence of a stable equilibrium and the boundedness of all solutions. For strictly dissipative systems, this guarantee extends to the global asymptotic stability of the equilibrium. Evaluation with real-world benchmark data demonstrates the sPHNN's ability to generalize from sparse data, outperforming the purely data-driven approach and avoiding instability issues. In addition, the model's potential for data-driven reduced order modeling is highlighted by training it on multi-physics simulation data to construct a surrogate model. When utilizing augmented dimensions, the stability constraint enables a safe and stable exploration of the added flexibility. While sPHNNs are confined to modeling globally stable systems, in their applicable domain, they promote robustness and physically plausible dynamics.