A long time ago, I did something very similar with an early release Magellan 
5000 board  (and a couple of other models of GPS cards) that did not implement 
the 1PPS signal.  I think the best time sync that I ever got was around 1 mS 
(most boards were around 10 mS).  

True 1PPS signals are implemented and synchronized in hardware.  The NMEA 
messages are generated in software.  Due to the vagaries of things like 
software interrupt latencies,  serial port bit synchronization, etc I doubt 
that you could ever get better than 1 serial bit time of accuracy out of any 
board and 10-100 bit times is more likely.  At 4800 bits/sec the jitter/offset 
of even 1 bit time would be huge at best.  Perfectly OK for things like clock 
displays,  but totally inadequate for things like oscillator locking.
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