Hubble Space Telescope Near-Infrared Camera and Multiobject Spectrometer (NICMOS) observations of six brown dwarf candidates in the Pleiades open cluster are presented. One of them, namely CFHT-Pl-18, is clearly resolved as a binary with an angular separation of 0.33". The very low density of contaminating background stars in our images and the photometry of the components support that this system is a physical binary rather than a chance projection. All of the available photometric and spectroscopic data indicate that the CFHT-Pl-18 system is likely a member of the Pleiades cluster, but a final confirmation will have to wait until lithium can be detected. Assuming cluster membership, we compare our NICMOS photometry with evolutionary models and find that the inclusion of the effects of dust grains is necessary for fitting the data. We estimate that the masses of the components are about 0.045 and 0.035 M_solar. The binary system has a projected separation of 42 AU (for a distance of 125 pc) that is common among stellar binaries.