I'm not sure why you’re struggling with incandescent/fluorescent @ ISO 1600. Are you shooting in Av mode? If not, try Av mode with aperture set to 2.0-2.8. If that's not enough, try 1.8. You should be able to get relatively sharp photos at a shutter speed of 1/60th or faster. I like at least 1/125th. My hands aren't super steady. You could use a tripod but I wouldn't let your shutter speed get any lower than 1/15th with 2 sec timer and make sure your wife and newborn are very still. If the baby comes during the day, let outdoor light into the room. With north style window light you should be able to shoot ISO800, f2.8-f4.0, 1/125th+.
As a last resort, if you don't already have one, buy an inexpensive swivel head flash. Here's the one I use:
http://www.ritzcamera.com/product/QTB9550CN.htm
Instead of blinding momma and the baby, simply point the flash over your shoulder at the wall-ceiling corner and bounce light onto momma and baby. I would use ISO800, f4.0, at 1/125th-1/250th. Turn up flash compensation +1 - +2 depending on how reflective the wall/ceiling is. This kind of flash is very unobtrusive and yields great results. That is.....if momma will agree to at least that.
I took up serious photography a while after my last of 3 boys was born but from what I can remember the hospital room was pretty well lit if you turn all the lights on. Baby may not like it after being in the dark for so long and you certainly don't want pictures of the new born crying.
My primo low light setup for 40D would be ISO1250 (optimum noise/sensitivity)
50mm 1.4 @ f2.0, very sharp, descent dof.
Shutter speed at 1/60th to 1/125th sec. (if higher, lower ISO)
Shoot in RAW + jpeg. If jpegs are good great, but RAW gives you more dynamic range and more ability to optimize shots later.
EV compensation of +.3. (shoot to the right) it helps noise if you need to lighten in post processing. If you need to darken in post this will really help cut the noise down. It will look better than ISO800 at 0EV!!
Try not to under expose. Boosting high ISO images in pp really exagerates the noise.
Hope this helps,
Chuck
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Blessed is the man who finds wisdom, the man who gains understanding,... (Prov 3:13 NIV)