In this work, we propose the use of Ground Penetrating Radar (GPR) for rover localization on Mars. Precise pose estimation is an important task for mobile robots exploring planetary surfaces, as they operate in GPS-denied environments. Although visual odometry provides accurate localization, it is computationally expensive and can fail in dim or high-contrast lighting. Wheel encoders can also provide odometry estimation, but are prone to slipping on the sandy terrain encountered on Mars. Although traditionally a scientific surveying sensor, GPR has been used on Earth for terrain classification and localization through subsurface feature matching. The Perseverance rover and the upcoming ExoMars rover have GPR sensors already equipped to aid in the search of water and mineral resources. We propose to leverage GPR to aid in Mars rover localization. Specifically, we develop a novel GPR-based deep learning model that predicts 1D relative pose translation. We fuse our GPR pose prediction method with inertial and wheel encoder data in a filtering framework to output rover localization. We perform experiments in a Mars analog environment and demonstrate that our GPR-based displacement predictions both outperform wheel encoders and improve multi-modal filtering estimates in high-slip environments. Lastly, we present the first dataset aimed at GPR-based localization in Mars analog environments, which will be made publicly available upon publication.