Determining the location of a tropical cyclone's (TC) surface circulation center -- "center-fixing" -- is a critical first step in the TC-forecasting process, affecting current and future estimates of track, intensity, and structure. Despite a recent increase in the number of automated center-fixing methods, only one such method (ARCHER-2) is operational, and its best performance is achieved when using microwave or scatterometer data, which are not available at every forecast cycle. We develop a deep-learning algorithm called GeoCenter; it relies only on geostationary IR satellite imagery, which is available for all TC basins at high frequency (10-15 min) and low latency (< 10 min) during both day and night. GeoCenter ingests an animation (time series) of IR images, including 10 channels at lag times up to 3 hours. The animation is centered at a "first guess" location, offset from the true TC-center location by 48 km on average and sometimes > 100 km; GeoCenter is tasked with correcting this offset. On an independent testing dataset, GeoCenter achieves a mean/median/RMS (root mean square) error of 26.9/23.3/32.0 km for all systems, 25.7/22.3/30.5 km for tropical systems, and 15.7/13.6/18.6 km for category-2--5 hurricanes. These values are similar to ARCHER-2 errors when microwave or scatterometer data are available, and better than ARCHER-2 errors when only IR data are available. GeoCenter also performs skillful uncertainty quantification (UQ), producing a well calibrated ensemble of 200 TC-center locations. Furthermore, all predictors used by GeoCenter are available in real time, which would make GeoCenter easy to implement operationally every 10-15 min.