This is really sort of cheating, but it could actually work, so here it goes.
As professorfish says,
It is theoretically possible, but I don't know how you could rip off the atmosphere and mantle.
Very true. Jupiter's atmosphere is about 5,000 km high, and comprises the majority of the planet's mass. That would be pretty hard for a civilization to get rid of. Fortunately, stars can do it for you - if you're willing to move the planet in question.
Hot Jupiters are large gas giants orbiting close to their home stars (0.015 AU and 0.5 AU, as opposed to Jupiter's average orbital radius of about 5 AU). Hot Jupiters are - well, hot because they are so close to the star, and so they suffer from other stellar effects. They most likely form farther out from the star - beyond the frost line - and come inwards, towards the star.
Hot Jupiters often experience strong stellar winds. These winds can, if they are strong enough, rip away the planet's atmosphere via hydrodynamic escape. If the process goes on long enough, the atmosphere can be entirely blown into space; the resulting body is known as a chthonian planet. These hypothetical objects would be similar to terrestrial planets in size. If you could introduce an atmosphere and other materials, you could begin to terraform one.
So a star can convert a gas giant to a terrestrial planet for you. The problem is that the planet has to already be very close to the star. Jupiter would have to have its orbit reduced to a very tiny size for this to be possible. To move it inward, you would have to induce an artificial migration. Perhaps you could attract it with another body - perturbing it just enough to make it begin to move inwards. Or maybe you could build a large object resembling a Dyson sphere in size, and use that to attract it inwards. Both options would be incredibly hard, but still within the realm of possibility.
You still have a problem: The chthonian planet is still not a great place for life. It is too close to the star to be inside its habitable zone, and it's way to hot. To get it further back, perhaps you could use the same mechanism you used to get it inwards: using the gravity of another body to change its orbit, a bit like planetary migration, although stability will, of course, be an issue. Once you have the planet where you want it, you can begin the process of terraforming it.