Hi, KT,
As you probably know, the 1903 was chambered for the Winchester automatic cartridge. It was made until about 1932, with about 126,000 made. I know of no source for a barrel of the original inside diameter (a little larger than the .22 LR), nor are there any liners available for the old .22 auto. A good gunsmith should be able to line the barrel for .22 LR, and make the other modifications necessary to get things to work with .22LR. Gun Parts Corp. has some parts for the Model 63, which was almost the same, but in .22 LR.
Problem is that the little gun has almost no value except as a collector's item and lining or replacing the barrel pretty well destroys that. It is an interesting problem.
Jim
