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Weird that it would happen on filling so makes me think it's an open on the sender's resistance circuit, (or as diabolical says, ruptured float maybe) . The first easy thing I would do is drive it enough to get the fuel level down and see it the gauge comes back to life. The other easy thing is to measure the resistance coming from the sender unit, while it's still in the tank. Should be ~ 5 or 10 Ω at full. The sender works by providing a variable resistance from about 10 Ω (Full) to about 110 Ω (Low on gauge). So your particular fault indicates an infinite (open circuit ) or abnormally high resistance
I would check the grounds on the car making sure they are clean. I know there's a strap from the muffler to the frame (thanks t_g_farrel) shown in the picture below. And you also want to make sure the connector to the sender is clean on both sides.
You will most likely want to remove and inspect the sender unit so know that when removing it, you should be very careful with the screws around the flange. They strip easily so make sure to use the perfect driver and be progressive and firm but do not force it.
