My only connections are GPS output [yellow wire from its supplied lead dress] to Arduino "D0" (and, of course, their Ground/common together.) Is D0 = RX on the Mega, too?
When I power up the arduino without the serial monitor it still goes, but when I push the Serial Monitor button in the IDE it resets the arduino, and then it's off to the races (data out each second).
And I do not have the GPS input (its blue wire) connected to D1.
I set it up again here on the bench, before posting just now, and it runs along without a hitch. One thing is, though, the output is the Decimal value of the ASCII code for the seconds digits (e.g. "48 48" for secs = 00, "48 57" for secs = 09).
*** The Arduino is running off the USB but the GPS is on a separate 5V. ***