Popular debiased estimation methods for causal inference -- such as augmented inverse propensity weighting and targeted maximum likelihood estimation -- enjoy desirable asymptotic properties like statistical efficiency and double robustness but they can produce unstable estimates when there is limited overlap between treatment and control, requiring additional assumptions or ad hoc adjustments in practice (e.g., truncating propensity scores). In contrast, simple plug-in estimators are stable but lack desirable asymptotic properties. We propose a novel debiasing approach that achieves the best of both worlds, producing stable plug-in estimates with desirable asymptotic properties. Our constrained learning framework solves for the best plug-in estimator under the constraint that the first-order error with respect to the plugged-in quantity is zero, and can leverage flexible model classes including neural networks and tree ensembles. In several experimental settings, including ones in which we handle text-based covariates by fine-tuning language models, our constrained learning-based estimator outperforms basic versions of one-step estimation and targeting in challenging settings with limited overlap between treatment and control, and performs similarly otherwise.