In this paper, we present a method for localizing a query image with respect to a precomputed 3D Gaussian Splatting (3DGS) scene representation. First, the method uses 3DGS to render a synthetic RGBD image at some initial pose estimate. Second, it establishes 2D-2D correspondences between the query image and this synthetic image. Third, it uses the depth map to lift the 2D-2D correspondences to 2D-3D correspondences and solves a perspective-n-point (PnP) problem to produce a final pose estimate. Results from evaluation across three existing datasets with 38 scenes and over 2,700 test images show that our method significantly reduces both inference time (by over two orders of magnitude, from more than 10 seconds to as fast as 0.1 seconds) and estimation error compared to baseline methods that use photometric loss minimization. Results also show that our method tolerates large errors in the initial pose estimate of up to 55° in rotation and 1.1 units in translation (normalized by scene scale), achieving final pose errors of less than 5° in rotation and 0.05 units in translation on 90% of images from the Synthetic NeRF and Mip-NeRF360 datasets and on 42% of images from the more challenging Tanks and Temples dataset.