  We report on sensitive sub-mm imaging observations of the prototype
Seyfert~2/starburst galaxy NGC 1068 at 850 $\mu $m and 450 $\mu $m using the
Submillimetre Common-User Bolometer Array (SCUBA) on the James Clerk Maxwell
Telescope (JCMT). We find clear evidence of dust emission associated with the
extended HI component which together with the very faint $ ^{12}$CO J=1--0
emission give a gas-to-dust ratio of $\rm M_{\rm gas}/M_{\rm dust} \sim
70-150$. This contrasts with the larger ratio $\rm M_{\rm gas}/M_{\rm dust}\sim
330$ estimated within a galactocentric radius of $r\leq 1.36$ kpc, where the
gas is mostly molecular and starburst activity occurs. The large gas-to-dust
ratio found for the starburst region is attributed to a systematic overestimate
of the molecular gas mass in starburst environments when the luminosity of the
$ ^{12}$CO J=1--0 line and a standard galactic conversion factor is used. On
the other hand sub-mm imaging proves to be a more powerful tool than
conventional CO imaging for revealing the properties of the diffuse $\rm H_2$
that coexists with HI. This molecular gas phase is characterized by low
densities ($\rm n(H_2)<10^3$ cm$ ^{-3}$), very faint emission from
sub-thermally excited CO, and contains more mass than HI, namely $\rm
M(H_2)/M(HI)\sim 5$.
