Extend a network from Pace 5268AC (potentially with Apple Airport)

I just got Internet 1000 and U-verse installed today. The speed is great... in the front of my house. In the back, it's really, really slow. This isn't a problem unique to the equipment—my old network had this problem too, but I was able to set that up in the middle of the house to counter that. AT&T recommended I set this up on the front room.

I'm okay with that, but now I need to figure out how to extend the network to the back of the house.

I would LOVE to do it with the Apple Airport Extreme Base Station (ME918LL/A) that I already have. I do have the ability to do this through ethernet, rather than just boosting Wifi signal.

Still, the only thing I've had luck doing so far is setting up the Airport to be it's own new network in the back of the house. Not my ideal solution because my devices don't automatically switch over. What I really want to do is extend the SAME network.

If I can't do it with my Airport, so be it, surely something exists that can. Any help would be greatly appreciated.

Thank you!