The Ruger 77UL is a good rifle, but with a 20" barrel I think I'd go with the 308Win. You'll probably need to bed the action/free float the barrel, and for sure will need a trigger job.
The Savage noted is also a good buy. They are very accurate.
For a carbine length rifle, you might also want to check out the Remington Model 7 - laminated stock, 20" barrel. Nice gun.
It's available in 308Win.
Personally, I wouldn't want a '06 that weighed less than 7lbs. before putting a scope, etc. on it.