Process for the continuous synthesis of methylamines by reaction of methanol and/or dimethyl ether with ammonia in the presence of a heterogeneous catalyst, wherein the catalyst used is a shaped body which comprises a microporous material and at least one organosilicon compound as binder and can be produced by a process comprising the steps (I) preparation of a mixture comprising the microporous material, the binder, a pasting agent and a solvent, (II) mixing and densification of the mixture, (III) shaping of the densified mixture to give a shaped body, (IV) drying of the shaped body and (V) calcination of the dried shaped body.