  At the highest stellar masses (log(\mstar) $\gtrsim$ 11.5 \msun), only a
small fraction of galaxies are disk-like and actively star-forming objects.
These so-called `super spirals' are ideal objects to better understand how
galaxy evolution proceeds and to extend our knowledge about the relation
between stars and gas to a higher stellar mass regime. We present new CO(1-0)
data for a sample of 46 super spirals and for 18 slightly lower-mass
(log(\mstar) $>$ 11.0 \msun ) galaxies with broad HI lines -- HI fast-rotators
(HI-FRs). We analyze their molecular gas mass, derived from CO, in relation to
their star formation rate (SFR) and stellar mass, and compare the results to
values and scaling relations derived from lower-mass galaxies. We confirm that
super spirals follow the same star-forming main sequence (SFMS) as lower-mass
galaxies. We find that they possess abundant molecular gas, which lies above
the extrapolation of the scaling relation with stellar mass derived from
lower-mass galaxies, but within the relation between \mmol/\mstar and the
distance to the SFMS. The molecular gas depletion time, \taudep = \mmol/SFR, is
higher than for lower-mass galaxies on the SFMS (\taudep = 9.30 $\pm$ 0.03,
compared to \taudep = 9.00 $\pm$ 0.02 for the comparison sample) and seems to
continue an increasing trend with stellar mass. HI-FR galaxies have an
atomic-to-molecular gas mass ratio that is in agreement with that of lower-mass
galaxies, indicating that the conversion from the atomic to molecular gas
proceeds in a similar way. We conclude that the availability of molecular gas
is a crucial factor to enable star formation to continue and that, if gas is
present, quenching is not a necessary destiny for high-mass galaxies. The
difference in gas depletion time suggests that the properties of the molecular
gas at high stellar masses are less favorable for star formation.
