I have this one - http://www.rackmountmart.com/rm1u/rm1002.htm
Yes, it's 1U and only 13 inches deep. This was my first PC (i'm a mac user) and I built it from parts. Building my first PC was hard enough, fitting it into 1U was quite a challenge. It is loud, but I use it for gigs and not so much in the studio. I guess that doesn't help you much. Some of the things I did learn, that may be worth mentioning....
I had no need for an optical drive and that saved me space and helped with airflow, ran cooler, less fan noise.
As with most computer PSU's there was about 2 miles worth of superfluous power cabling and plugs, after cutting that stuff out I had much better airflow, ran cooler, less fan noise.
I drilled 30 or so holes above the CPU. Better air flow, ran cooler, less fan noise.
Both the 2nd and 3rd tricks I rememberred from when a certain boss of mine 'modified' a certain noisy console that had respiratory issues inherent to it's design.