I'm not an "expert" but I'd vote (1) main relay or (2) ignition switch. Something is definitely cutting power to or from the position II wire on the ignition.
Next time it dies try fiddling w/ the key to see if it causes anything to come to life (in position II). Older Civics were plagued by bad solder joints in the main relay with similar (not identical) issues but I haven't heard of it being a problem with 98 era cars.
Not surprising that the starter will still turn since this is a separate/dedicated wire on the ignition switch (I believe). Checking the wiring diagram on the FSM would confirm this. And this would isolate the circuit that you're having problems with -- along w/ Dick's recommendations too (lighting, radio, etc)
_________________
Mike Whitney
whit32@gmail.com, 919-454-5445
V10, V8, V8t, I6, I6, V6, F4t, I4, I4, I4, I4, I2, 1, 1