The thermal-neutron fission cross section of $^{235}\mathrm{U}^{\mathrm{m}}$ has been measured relative to the ground-state cross section. A rapid radiochemical separation procedure was developed to provide sizeable (${10}^{10}$ to ${10}^{11}$ atom) samples that were reasonably free of the parent $^{239}\mathrm{Pu}$. From a series of eight measurements, the value of 1.42\ifmmode\pm\else\textpm\fi{}0.04 was obtained for the ratio ${\ensuremath{\sigma}}_{m}$/${\ensuremath{\sigma}}_{g}$.