| M14man, you're close on the period of manufacture. The Serial number 1365121 appears to have been a 1930 number (1338406‑1369760). You're absolutely correct about the re-barreling during World War II. Almost all of America's existing small arms went through the arsenal rehabilitation program that included new parts when and where needed.
ttaylor, you have one of America's premier service rifles manufactured from 1903 until 1930/31 by Springfield Armory and to a lesser extent; Rock Island Arsenal. Remington Arms made the same ’03 design from 1939 until 1941. A revised design to facilitate ease of manufacturing called the M-1903A3 was manufactured from 1942 until 1944 by L.C. Smith-Corona and Remington Arms Company.
Your ‘03 was too late to have served in World War I, but it certainly saw service during World War II. ‘03s and 03-A3s were designated “substitute standard issue” service rifles during a time when the M1 Garand service rifle was the “standard issue.” During World War II, there were never enough M1s, so the ‘03s and ‘03-A3s were issued when and where needed. They served primarily in the South Pacific in the Solomon Islands Campaign and in North Africa, and continued to be found at the front and in reserve throughout the war. | 
