We examine the modes of a scalar field in de Sitter space and construct quantum two-point functions. These are then used to compute a finite stress tensor by the technique of covariant point-splitting. We propose a renormalization ansatz based on the DeWitt-Schwinger expansion, and show that this removes all ambiguities previously present in point-splitting regularization. The results agree in detail with previous work by dimensional regularization, and give rise to an anomalous trace with the conventional coefficient. We describe how our treatment may be extended to more general situations.