Do not put her down. Separate her from the others, soak her little behind to get the poop off, then trim the feathers away from all her back under side. She will continue to have a mess of a behind while the uterus is prolapsed and by trimming away the feathers she'll keep much cleaner. Still, soak her every now and then, apply some triple antibiotic to lower the risk of infection. Feed her yogurt with her usual feed to boost her bacterial resistance.
My little black naked neck had a terribly prolapsed uterus at the end of last winter and she is now running around back out there with the little flock happy as the day is long.
Hens do not die from prolapsed uterus. They can die from infection and certainly can die if the other hens get the idea to pick at the red area (and cause bleeding and--infection).
I got tips to keep her food and light limited to slow her laying down, but she never stopped for a day. She lay right through it. For all the times I pushed her uterus in she fought like crazy and pushed it out farther. After three months of separation and only small improvement, I let her rejoin the flock during the day, then at night. A month or so after that her uterus popped right back in and she hasn't had a problem since.
The condition never seemed to bother her. (My baths, pushing on her behind, shutting her away from her pals, the cool dark basement--THESE things bugged her. Never the red internal organ hanging out her behind.)
If you have any other questions about how my Hera pulled through, feel free to ask. Keep the faith!