We have used the Near-Infrared Camera and Multi-Object Spectrometer NIC1 camera on the Hubble Space Telescope to obtain high angular resolution images of 52 ultracool dwarfs in the immediate solar neighborhood. Nine systems are resolved as binary, with component separations from 1.5 to 15 AU. Based on current theoretical models and empirical bolometric corrections, all systems have components with similar luminosities and, consequently, high mass ratios, q>0.8. Limiting analysis to L dwarfs within 20 pc, the observed binary fraction is 12%+7-3. Applying Bayesian analysis to our data set, we derive a mass-ratio distribution that peaks strongly at unity. Modeling the semimajor axis distribution as a logarithmic Gaussian, the best fit is centered at loga0=0.8 AU (~6.3 AU), with a (logarithmic) width of +/-0.3. The current data are consistent with an overall binary frequency of ~24%.Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.