[JamesP81]

It depends on if we mean 390 meters or 390 miles. If its meters, then I suggest a layered defense.

We will need a robotic spacecraft capable of moving to an asteroid. Instead of carrying a robot to study it, it needs to carry the most powerful hydrogen bomb as can be lifted by the vehicle. Send on bomb directly towards it and detonate it short of the rock, with intent to deflect. Also, send several bombs towards the paths the rock may take after being deflected by the first blast, so you'll have one in place to detonate no matter which way it goes. Simultaneously send numerous bombs in a dispersal from the Earth towards the asteroid, and do this in three or more layers. Detonate these in sequence to vaporize the smaller particles as they pass near these lines of bombs. By the time whats left of the asteroid gets close to earth, hopefully what's left of it can be vaporized by earth's atmopshere.

[z3n]

"We will need a robotic spacecraft capable of moving to an asteroid"

This is feasable given enough for-warning and time to plan (which we obviously have). We simply preset large amounts of solid rocket fuel into orbit (or other type of fuel to be determined), and then launch a mission to put the robot to link up in orbit with it. From there, the robot will have enough fuel to rendezvous with the NEO many months (or years) before impact, and with enough fuel left over (planned this way) mount the object and continuously fire it's rockets against the object in a direction perpendicular to it's trajectory (or however it should be engineered so that the tragectory is altered sufficiently).