A rocket that moves upward from earth's surface at escape velocity will?

Escape the earth's gravitational pull and continue out into space.

However, a rocket does not need to be launched at the escape velocity as it can continue to accelerate as it climbs. A gun projectile would need to be fired with the escape velocity.

In a perfect system with only the projectile and the Earth:

If the projectile is fired with the exact escape velocity it will travel to infinity away from the Earth. Upon reaching infinitely far away from Earth the projectile would have zero velocity. All of its kinetic energy (movement) would be transferred to potential energy.