We present CO J=1-->0 observations (lambda=2.6 mm) of 10 polar-ring galaxies, chosen from the Polar Ring Catalogue; we infer masses of H_2 ranging from 7x10^7 to 2x10^10 Msolar, with an average of about 1x10^9 Msolar. These H_2 masses are greater than the average molecular mass of an early-type galaxy; we previously found similar results for a sample of minor-axis dust-lane ellipticals. In the cases where we can estimate the gas mass in the polar ring, including the H i masses from literature, they are high enough to allow self-gravitation to stabilize the rings. This means that the ages of the rings may be >=1 Gyr. Indeed, the gas masses are often greater than those of most dwarf galaxies: this would make it unlikely that the polar rings r...