I just happen to have a (nearly) one inch long piece of a stainless barrel that was cut off a factory .300 Weatherby. I threw it on the E-scale and the .933 inch long stub weighed 386 grains. If anyone wants to do the conversion to ounces ....a full 2 inches of this barrel would have weighed approximately 778 grains.
The barrel was a #2 and .600" at the original muzzle (26").

S.B.