With a full gig of video memory available, you almost undoubtedly will be able to achieve 1920x1080 resolution with that laptop and that monitor.
I know that on a considerably lesser video arrangement (I forget the chip, but it's 256M of shared video memory), I have no problems driving 1680x1050 on my external monitor -- while also simultaneously driving the built-in 1280x800 display. Now, that's less than your desired resolution, but it's also with 1/4 the video memory.
Now, there's no guarantee you'll be able to play super-finely-detailed games at what amounts to 1080p on that monitor -- but you should be able to use it as a very effective secondary display.
Also, if you were to enable mirroring from your laptop to said display, the secondary display will be limited to your laptop's native 1366x768 (roughly 720P) resolution.