I've seen hedges where people have trained climbers up through the dead sections to disguise them - or, of course, take out the dead branches and replant - it'll take a while to fill the gaps, though.

It's best to take only the dead leaves off the dead branches. Spray the dead branches with a fungicide, you can buy this in a hand held sprayer. The old branches will then hold the new growth and you won't have a hole in the hedge. Treat around the roots with Armillatox.