Mars today has no core dynamo magnetic field. However, the discovery of remanent magnetization in Martian meteorites and intense crustal magnetization suggests that Mars once had a global field. Here we present high resolution maps of the magnetic field of Martian meteorite ALH 84001. These maps are the most sensitive yet quantitative study of natural remanent magnetization (with resolved anomalies as weak as 1 × 10−14 Am2). ALH 84001 likely contains a 4 billion year old (Ga) thermoremanence partially overprinted by one or more poorly understood secondary components. Our data suggest that the paleointensity of the local paleofield was within an order of magnitude of that of the present-day Earth. If this field were global in extent, it should have played a key role in Martian atmospheric and climatic evolution. However, it is still too weak to easily explain the intensity of Martian crustal paleomagnetic anomalies.