Forced to manage a chronically negative subordinate? Tips on how to do it (and stay sane)

Employees come in many different forms. They can be cheerful, dedicated and motivated, or permanently negative, reluctant and passive. Unfortunately, one negative subordinate can ruin all efforts to motivate a team as they will infect their colleagues with their negativity. How to manage a negative colleague who is constantly complaining, sees everything as a problem and spreads a bad mood around the workplace?