Images from the Faint Object Camera (FOC) on the Hubble Space Telescope (HST) are used to spatially separate the two giants of Capella (α Aurigae; HD 34029) for the first time at ultraviolet wavelengths. The images were obtained with broadband filters that isolate the wavelength regions 2500-3000 Å and 1300-1500 Å. The cool G8 giant is found to be weaker than the hot G1 giant by factors of around 4 and 17, respectively, in these bands. The latter factor is largely due to the much stronger G1 continuum at short wavelengths. No evidence is found for material lying between the two stars in the images. In addition, the objective prisms of the FOC were used to obtain low-resolution spectra from 1200 to 3000 Å, allowing individual emission lines from each star to be spatially separated. Cool-to-hot star ratios for the emission lines H I Lyα, O I λ1305, Si II λ1816, C II λ1335, He II λ1640, and Si IV λ1393 are presented, showing that the cool giant is weaker than the hot giant by factors of 5-10 in these lines. The O I emission is only a factor of 2.5 weaker in the cool giant, most probably resulting from fluorescence in the extended atmosphere of the cool giant. The line ratios are compared with values derived from International Ultraviolet Explorer and HST/Goddard High Resolution Spectrograph spectra, which could separate the stars spectrally but not spatially. Reasonable agreement is found although the FOC ratios generally imply lower contributions from the cool giant. Based on observations with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.