WAUW, now it works perfectly 
I'm using this code:
void setup() {
Serial.begin(4800);
}
void loop() {
if (Serial.available()) {
char hardSerialChar = Serial.read();
Serial.print(hardSerialChar, BYTE);
}
}
The code without the if Serial.available inserted alot of "-1" in the NMEA-sentenses (hmm, one could actually figure out the cycle-speed of the 'void loop' function by looking at how many "-1" are for each NMEA-letter, and knowing that the baud rate of the GPS is 4800)
I had some issues right up till now. Using the power supply from the breadboard to support my GPS made it not turn on (which I didn't notice). Using the ground from the breadboard and the power line from the original GPS-cable also didn't work - the GPS turned on but couldn't communicate anything. Finally I added a wire from the ground on the breadboard to the ground-jack on the original GPS cable going to the computer. THEN it worked. PHEW!! I was getting really frustrated.
And yes: DON'T let data go into RX when uploading new code to the Arduino.