The Smith Cloud is a high-velocity cloud with an orbit suggesting it has made at least one passage through the Milky Way disk. A magnetic field found around this cloud has been thought to provide extra stability as it passes through the Galactic halo. We use the Karl G. Jansky Very Large Array to measure Faraday rotation measures (RMs) toward 1105 extragalactic background point sources behind and next to the Smith Cloud to constrain the detailed geometry and strength of its magnetic field. The RM pattern across the cloud gives the detailed morphology of the magnetic field structure, which indicates a field draped over the ionized gas and compressed at the head of the cloud. We constrain the peak line-of-sight magnetic field strength to >~+5{mu}G and create a model of the magnetic field to demonstrate that a draped configuration can qualitatively explain the morphology of the observed RMs.