We introduce Sparse Symplectically Integrated Neural Networks (SSINNs), a novel model for learning Hamiltonian dynamical systems from data. SSINNs combine fourth-order symplectic integration with a learned parameterization of the Hamiltonian obtained using sparse regression through a mathematically elegant function space. This allows for interpretable models that incorporate symplectic inductive biases and have low memory requirements. We evaluate SSINNs on four classical Hamiltonian dynamical problems: the Hénon-Heiles system, nonlinearly coupled oscillators, a multi-particle mass-spring system, and a pendulum system. Our results demonstrate promise in both system prediction and conservation of energy, often outperforming the current state-of-the-art black-box prediction techniques by an order of magnitude. Further, SSINNs successfully converge to true governing equations from highly limited and noisy data, demonstrating potential applicability in the discovery of new physical governing equations.