Considering that a 24" varmint taper barrel (1.055" at the shank and 0.835" at the muzzle) with a .224" bore weighs 4.5 lbs, here you go some calculated weight estimates for a bull barrel (no taper) with a 6mm/.243" bore:

26" - 6 lb 1 oz
28" - 6 lb 9 oz

These are just estimates. They could be off by a couple of ounces for I disregarded any differences in chamber volumes. Also there could be differences in steel density.