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We report on sensitive sub-mm imaging observations of the prototypeSeyfert~2/starburst galaxy NGC 1068 at 850 $\mu $m and 450 $\mu $m using theSubmillimetre Common-User Bolometer Array (SCUBA) on the James Clerk MaxwellTelescope (JCMT). We find clear evidence of dust emission associated with theextended HI component which together with the very faint $ ^{12}$CO J=1--0emission give a gas-to-dust ratio of $\rm M_{\rm gas}/M_{\rm dust} \sim70-150$. This contrasts with the larger ratio $\rm M_{\rm gas}/M_{\rm dust}\sim330$ estimated within a galactocentric radius of $r\leq 1.36$ kpc, where thegas is mostly molecular and starburst activity occurs. The large gas-to-dustratio found for the starburst region is attributed to a systematic overestimateof 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. Onthe other hand sub-mm imaging proves to be a more powerful tool thanconventional CO imaging for revealing the properties of the diffuse $\rm H_2$that coexists with HI. This molecular gas phase is characterized by lowdensities ($\rm n(H_2)<10^3$ cm$ ^{-3}$), very faint emission fromsub-thermally excited CO, and contains more mass than HI, namely $\rmM(H_2)/M(HI)\sim 5$.Comment: Accepted for publication in the Astrophysical Journal Letter

Extended Dust Emission and Atomic Hydrogen: A Reservoir of Diffuse H[TINF]2[/TINF] in NGC 1068