Process for producing a supported catalyst which comprises at least 75% by weight of Al.sub.2O.sub.3, whose proportion of Al.sub.2O.sub.3 in the delta or theta modification is, based on the proportion of Al.sub.2O.sub.3, at least 1% and which comprises a rhenium compound and, if appropriate, a promoter as active component (A), which comprises a) converting a customary support (S) which comprises at least 75% by weight of Al.sub.2O.sub.3 and to which a promoter may, if appropriate, have been applied is converted into a modified support (S) whose proportion of Al.sub.2O.sub.3 in the delta or theta modification is, based on the proportion of Al.sub.2O.sub.3, at least 1% by calcining the customary support (S) at a temperature of from 750 to 1100.degree.C., b) producing a supported catalyst precursor from the modified support (S) by applying the active component (A) comprising the rhenium compound to the modified support (S) and c) calcining the supported catalyst precursor at a temperature of from 500 to 750.degree. C.