We have observed millimeter wavelength spectra from the narrow-line radio galaxy Cygnus A in a search for CO(1-0) absorption from a hypothesized dusty molecular torus surrounding the central nucleus. We find no evidence for CO absorption. This result may be consistent with a model in which the torus is composed of small, opaque clouds with a covering factor of roughly unity along lines of sight that pass through the torus; the clouds must be small compared to the size of the compact radio source. If the CO is thermally excited, the space between the clouds must be very clear of molecular gas, perhaps implausibly so. However, it has been suggested to us [M. Rees, private communication (1993)] that the presence of the strong compact radio source could result in a very high rotation temperature for molecules in a compact torus, possibly accounting for the lack of absorption. In the absence of such nonthermal excitation effects, our observations argue against tori composed of diffuse molecular gas, or ones where the clouds have a covering factor much greater than 1 through the torus.