Thanks for your response. I need to learn how to use the forum's search function better...
I ordered two modules from the same supplier, both showing the same behaviour. They look exactly the same as the module in the thread you found. Either this supplier is sending out bad modules, or I am breaking them. I have the arduino's 5 V and ground directly on Vcc and GND. TXD goes to RXpin (14) and RXD to TXpin (15).
I will order one from another supplier, see if that solves the problem.
Kind regards.