We have measured the Faraday rotation toward a large sample of polarized radio sources behind the Large Magellanic Cloud (LMC) to determine the structure of this galaxy's magnetic field. The magnetic field of the LMC consists of a coherent axisymmetric spiral of field strength approximately 1 microgauss. Strong fluctuations in the magnetic field are also seen on small (<0.5 parsec) and large (approximately 100 parsecs) scales. The large bursts of recent star formation and supernova activity in the LMC argue against standard dynamo theory, adding to the growing evidence for rapid field amplification in galaxies.