Your description is rather confusing, can you draw a diagram?
Normally one connects the TX (transmit/output pin) of one device to the RX (receive/input pin) of the other device. I know there can be naming issues with DTE/DCE, but always go with the physics. I know the FT232BM (DLP-TXRX) works just fine with the STM32. http://www.dlpdesign.com/dlptxrx-v12-ds.pdf
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''I know it is weird, but really I have to connect RX to RX and TX to TX''
Actually, that's not weird at all - that's perfectly normal when one device is a DCE (modem) and the other is a DTE. This is, in fact, the way RS232 was designed to work! In this case, you should think of ''RX'' and TX'' as circuit names - not pin names: The RX cicuit carries the data thas has been received from the comms medium; The TX circuit carries the data thas is to be transmitted to the comms medium. Thus the TX pin will be an output from the DTE, and an input to the DCE; the RX pin will be an input to the DTE, and an output from the DCE. That's why clive1 suggested that you look at the ''physics'' rather than the names: an output pin must always drive an input pin!
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