I haven't tried it, but I don't think it works. If your car radio can't receive Bluetooth, there are Bluetooth receivers that plug into the aux socket, e.g. something like this:
LogiLink BT0022 - Bluetooth Audio Receiver

It does not depend on the stick, but on the car radio. Your radio should be compatible with the CSR chipset, support the Bluetooth stack (the high speed version of Bluetooth) and the corresponding audio profile A2DP. These are all software components that must be present in your radio... Look it up in the manual...
What you need is a "Bluetooth Audio Adapter" which can be found under "Audio + Hifi".

so that the radio can play mobile phone music? It depends on whether the radio supports it.

Conversely, if you connect the module to your laptop, you can already use it to broadcast to your car radio.