You'll need some sort of video capture. Ideally HDMI for quality, but then I think you'll run into HDCP issues if the Uverse box requires it (which I suspect it does). Component would be the next best option and still lets you do limited HD.
The Blackmagic Intensity Pro has HDMI and is surprisingly cheap ($200), but I doubt it handles HDCP.
An HDfury would solve the HDCP problem, but is analog output only so you'd need a capture card with RGBHV or component input.