Cut the dead growth off to the base - there will soon be new shoots appearing.

Mint is herbaceous and will regrow. However, watch out for the creeping stems as these will sprout away from the original clump. This isn't a problem as you can dig out any wayward bits or use them for propagation material. If you put a small cloche (large fizzy drink bottle with the bottom cut out etc) over a piece it may be encouraged to leaf up earlier than the rest, giving you early shoots to pick.