  Using the IRAM 30m telescope, we have surveyed an unbiased sample of 47
nearby spiral galaxies of very late (Scd-Sm) Hubble-type for emission in the
CO(1-0) and CO(2-1) lines. The sensitivity of our data (a few mK) allows
detection of about 60% of our sample in at least one of the CO lines. The
median detected H2 mass is $1.4\times 10^7 \msun$ within the central few kpc,
assuming a standard conversion factor. We use the measured line intensities to
complement existing studies of the molecular gas content of spiral galaxies as
a function of Hubble-type and to significantly improve the statistical
significance of such studies at the late end of the spiral sequence. We find
that the latest-type spirals closely follow the correlation between molecular
gas content and galaxy luminosity established for earlier Hubble types. The
molecular gas in late-type galaxies seems to be less centrally concentrated
than in earlier types. We use Hubble Space Telescope optical images to
correlate the molecular gas mass to the properties of the central galaxy disk
and the compact star cluster that occupies the nucleus of most late-type
spirals. There is no clear correlation between the luminosity of the nuclear
star cluster and the molecular gas mass, although the CO detection rate is
highest for the brightest clusters. It appears that the central surface
brightness of the stellar disk is an important parameter for the amount of
molecular gas at the galaxy center. Whether stellar bars play a critical role
for the gas dynamics remains unclear, in part because of uncertainties in the
morphological classifications of our sample.
