I have used multi-epoch astrometry from the Wide-field Infrared Survey Explorer to perform a search for a distant companion to the Sun via its parallactic motion. I have not found an object of this kind down to W2 = 14.5. This limit corresponds to analogs of Saturn and Jupiter at 28,000 and 82,000 AU, respectively, according to models of the Jovian planets by Fortney and coworkers. Models of brown dwarfs by Burrows and coworkers predict fainter fluxes at a given mass for the age of the solar system, producing a closer distance limit of 26,000 AU for a Jupiter-mass brown dwarf. These constraints exclude most combinations of mass and separation at which a solar companion has been suggested to exist by various studies over the years. Based on data from the Wide-field Infrared Survey Explorer and the Two Micron All-Sky Survey.