In a process for distilling polymerization-prone compounds from liquid mixtures of matter which, apart from the readily polymerizable compound, also contain other components and also polymerization inhibitors, the evaporator is operated in forced circulation and at a pressure which makes possible superheating of the liquid with respect to its boiling point at the pressure within the distillation column. This is achieved by transporting the superheated liquid against a flow restrictor, e.g. an orifice plate. The liquid which is expanded and partially evaporated on passage through the flow restrictor is returned to the distillation column. In this manner the formation of an open vapour space in the evaporator circuit after heating downstream of the heat exchanger is avoided in that, in the design of the evaporator of the prior art, formation of polymer deposits frequently occurs.