Waterman;
I did that exact same conversion several years ago. I opted for a new barrel. I bought a Shilen barrel from Brownell's. If you do as I did, and choose a stock barrel contour the barrels are quite inexpensive and they are "Match Grade". I had a match chamber cut and took the extractor to a welder who welded it up to be recut to .22 rimfire. You don't even have to move the firing pin. The original breech block will be just fine.

It has made a very satisfactory rifle. The Stevens 44 makes one of the best .22 rimfires you could choose. The "rocking block" will allow you to use a full match chamber without any problems as the block will seat the cartridge just fine. My chamber engraves the bullet almost to the case and it is a pleasure to use. It also shoots extremely well ("X" ring accuracy on the 50 yard NRA target at fifty yards).

YMMV

Good luck!
Dale53