Are there any other fish? If so, they will keep the parisite alive so that it would be pointless to remove the one fish, treat it, then return it. If you treat (copper or hypo) you need to treat ALL the fish.
Once there are no fish whatsoever in the display tank, it should stay fishless for 6 + weeks to be sure the ick dies out in that tank.
Do not use Copper and hypo together. I don't recall the exact risk, but IIRC doing both at the same time is very likely to kill the fish (serious internal organ damage IIRC).
FWIW, lots of people have ick in their tanks but just "manage" it by keeping stress low and feeding well. Fish do have some natural immunity, so if they are strong and healthy they will often fight the infection off for the most part. The downside is that whenever there is new stress in the tank (like adding a new fish) the ick is likely to re-appear. "managment" is an option.
If you do treat with hypo, you MUST test the salinity accurately (use a refractometer, NOT a swing arm hydrometer) and very often. Just a little evaporation can bring the SG up enough to make the treatment ineffective. The usual reccomendation is to keep the SG at 1.009. Personally I've had good luck keeping it at no higher than 1.008. Keep it at whichever hypo level you go with for at least 6 weeks, plus at least a few days to bring the SG down, and at least a week to bring it back up (it's much more important to bring it back up slowly, fish will usually tolerate a fairly rapid drop but not a rapid rise.)
If you treat with copper, the treatment only takes around 2 weeks, but the display will still need to stay fishless for 6 or more weeks. With copper, it's critical to test daily and keep the Cu at a theraputic level ALL the time. Definatly do your homework on this, there are different kinds of copper meds, and different types of test. I don't know enough to advise you on the specifics here, but be careful and make sure you know what your doing before starting.
Both Copper and Hypo can have risks. "management" also has it's risks. No one solution/approach is right for everyone.
Oh yea, Welcome to the club