Process for preparing a cyclic tertiary methylamine of the formula I where A is a C4-alkylene group, a C5-alkylene group or a —(CH2)2—B—(CH2)2— group, where B is oxygen (O) or an N—R1 radical and R1 is C1-C5-alkyl, aryl or C5-C7-cycloalkyl, wherein an amino alcohol II from the group consisting of 1,4-aminobutanol, 1,5-aminopentanol, aminodiglycol (ADG) or aminoethylethanolamine of the formula IIa where R1 is as defined above or is hydrogen (H), in which case R1═CH3 in the amine I, is reacted with methanol in a reactor at a temperature in the range from 150 to 270° C. in the liquid phase in the presence of a copper-comprising heterogeneous catalyst.