Don't know what your skill level is, but if you don't have some good guitar electronics repair knowledge or expirience, I would most strongly suggest that you have a qualified guitar tech check for the problem.That means doing some research and finding someone in your area with a solid reputation for working on vintage instruments, not some music store hack "tech". Don't know what you paid for it,but you have a pretty valuable guitar,worth many thousands of dollars,that needs to repaired correctly and professionally in order to retain it's value. Unless you know what you're doing, I wouldn't suggest working on this guitar yourself. Hollow body guitars are a bear to work on as far as removing the electronics and getting them back in. If you don't have the experience doing this,you can easily cause further damage to the electronics and the finish. This guitar also has the fairly complicated stereo/Varitone wiring circuit. All cautions being expressed, the first thing to do is to test the pickup to see if it is indeed dead. You would have to pull the pickup and test it with a multimeter. The meter needs to be set at 20k ohms. Each probe would be connected to one of the pickup leads and you should get a reading,for a Gibson pickup of that era,of somewhere in the 7-8 K ohms resistance range.If it reads zero,the pickup is indeed dead. In that case, your option would be to find and buy a used, vintage correct pickup, or have yours rewound by a top notch winder,such as Jason Lollar of Lindy Fralin, who can restore the pickup to original specs. Although it's not entirely impossible,it would be very rare as pickups almost never go dead on their own. If the pickup reads OK,the next suspect would be one of the switches. This is the most common cause of a pickup not working. The switch can be either physically broken or the contacts can be corroded and need to be cleaned with a good and proper electrical contact cleaner like DeoxIt.In order to do this right,you would have to get the switches out of the body.Never use WD-40 or some other similar substance for this.Contrary to what a lot of people believe, it is not the proper thing to use.A solder break can be a possibility,but that wouldn't really occur unless someone had monkeyed around inside the guitar and managed to break something.Checking for that would require taking out and putting the electronics back in, and like I said, on a hollow body that is not something to do unless you know what you're doing. Could also be one of the pots connected to that pickup. That may also be a matter of the pot being worn out,or also just needing some contact cleaner sprayed into it to clean out any corrosion. Just weigh your options carefully and really consider whether you want to tackle this or have a qualified person do
it. Those are beautiful guitars. A true classic.
Which version do you have?
Edit- Don't know if you're aware of the value,but this 1968 vibrato equipped one in excellent condition sold for $8595. Check out the link below.
Also,here's some great info on the Gibson ES-355 models.