A magnitude-limited survey of classical Cepheids brighter than 8th mag has been carried out to search for hot main-sequence companions. Spectra of 76 stars were obtained with the International Ultraviolet Explorer (IUE) satellite in the 2000-3200 angstrom region. They were compared with the spectra of nonvariable supergiants and also the single Cepheid delta-Cep to search for excess flux at 2500 angstrom from possible companions. Companions typically A1V or earlier were detected, although individual detection limits are given. Photometric companions were found for 21% of the sample. When the Cepheids known to be binary from either orbital motion or spectra in the 1200-2000 angstrom region are included, the percentage of companions rises to 29%. If a statistical correction from stars with orbital motion is included, 34% have companions. This percentage is compared with that found by Abt, Gomez, and Levy for B2-B5 main-sequence stars. If only systems with periods longer than a year and separations less-than-or-equal-to 30" (consistent with the IUE aperture) are considered, only 18% of the B stars will become Cepheids with companions. The difference between this fraction and the observed fraction of Cepheids with companions is probably due to undetected companions with periods longer than a year in the B-star sample, although there are uncertainties due to close binary evolution.