Thanks for the answers everybody,
TomGeorge:
Hi,
Can I suggestSerial.begin(115200);
GPS_Serial.begin(Baud_Rate);
And leave the IDE monitor at 115200. It doesn't have to be the same as the GPS baudrate. Just a thought. Tom... :)
Yes - I tried it, and in general it works.
I establish the connection between the Arduino to PC @115200 bps, and leave is stable. Just changing the baud rate between the Arduino and GPS.
One problem that I see - sometimes, when the baudrate of Arduino to PC is 115200, and GPS to Arduino is 4800 - I send a command to GPS and it goes gibberish - not accepted. It doesn't happen all the time, just sometimes.
Regarding connecting the GPS to pins 0,1 - I just wonder - aren't those the pins of the main serial connection of the UNO? Parallel to the connector to USB?
I was busy with other things but will try today to implement it with NeoSWSerial.
BTW the GPS is CSR SiRFStar V.