If you're living in Orson Scott Card's (or Ursula K. Le Guin's) universe, you could simply use an ansible relay.

For those not familiar with the Ender novels, the ansible is a clever little device to allow instantaneous communication between any two points in the universe. When it was first introduced in Ender's Game, there was no explanation as to how it worked; we were only told that humans learned of the existence of such technology from an alien life form that used it, and once they knew it was possible reverse engineering it was just a matter of time. By the time the next novel, Speaker for the Dead, takes place, ansibles are widespread and essential for communication among the interplanetary human colonies.

Mind you, sending spaceships faster than light is still impossible; the ansible only allows transmission of data. The idea is based on the idea of quantum entanglement, where two subatomic particles are "entangled" in such a way that each one is aware of the other's state, no matter how far they're removed from each other. Alter the state of one particle, and the other responds in the same manner, even if they're light years apart. Sending information encoded in binary through such a device is just a matter of switching states back and forth over and over again.

Unfortunately, there are real and quantifyable reasons why an ansible can't work in the real world, most of them involving a working knowledge of quantum mechanics. The proof of this is left to the reader as an exercise.