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 M _ { gas } / M _ { dust } \sim 70 - 150 . This contrasts with the larger ratio M _ { gas } / M _ { 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 H _ { 2 } that coexists with HI . This molecular gas phase is characterized by low densities ( n ( H _ { 2 } ) < 10 ^ { 3 } cm ^ { -3 } ) , very faint emission from sub-thermally excited CO , and contains more mass than HI , namely M ( H _ { 2 } ) / M ( HI ) \sim 5 .