When my xbox is not RRODed, I use one of these:
AverMedia AverTV HD DVR PCI-E Video Capture HDMI S-Video Digital Recording - AVERMEDIA - MTVHDDVRR
I also use it for VHS tapes.
It does not do 1080p though, just 720P and 1080i but I am not bothered by that. I should also say that I have not noticed any lag with this card.
Also looks like avermedia have a couple other cards out that say 1080p but I do not know anything about them.