Which alternator setup do you have Frank? Is it from the '93, or is it from the Bronco's original accessory setup? Or something different still?
If you're not sure, a few pics will tell us the story.
What year is your Bronco?
The main ground cable from the battery should not go straight to the frame. It should go directly to the engine block. And the nearer it is to the starter motor, the better.
Is the wiring original? Or has it been re-wired?
A common issue with stock wiring if there have been other electrical problems (known or unknown) would be a blown fusible link in the main Black wire. This is a bit of under-sized wire that is there to protect the overall system from bad things and is used instead of a fuse or circuit-breaker.
This is the larger Black wire (could have a red stripe, could have a yellow stripe, or might not have any stripe) and is attached to the starter relay/solenoid on the same stud as the main battery cable. You might be able to inspect it for being soft or swollen up from being melted inside. Check it out and let us know what you find.
A bad starter relay can't in any way keep the battery from charging. Period, end of story.
If it's wired properly, the charge wire and the positive battery cable are in physical contact with each other. If the alternator is charging and the output is reaching the end of the wire at the starter relay, there is nothing other than a defect in one of those two wires that can block current from getting from the alternator to the battery.
The physical contact guarantees that nothing else on the vehicle is at fault.
Let us know what kind of charging and wiring setup you have and we can go from there.
Paul