I have my eye on a new, dusty No.1A in .280 Remington with pretty wood, for $825.00 ( before I perform my Lebanese rug trader ju jitsu ).
They don't weigh as much as many bolt action rifles, and less than one with a moderator and Hubble scope.
They do have a fore end hanger which needs some adjusting. The last one I did on a friend's 7mm Rem Mag 1B shrunk the groups from 1.5 inches to .75 inches.
I had a No. 1 Light weight in 7x57 Mauser which was a nice rifle. A friend and I spotted it and we bought it together. He still has the rifle and loves it. I got the Swaro scope off it.
You have to like retro stuff, like Mausers, Model 70s, 1903 Springfields, etc. If you want a more accurate rifle out of the box, the 1885 Winchester, by Winchester, or the Browning 1885, or Browning B78 are the thing - all made by Miroku. But they are even heavier. My B78 in .30-06 with a 26-inch tapered octagon barrel is 8.25 lbs, then there is the Leupold base and ring set, and the scope, which bring it to 10 lbs.
The 1885 Low Wall in a .223 Rem, .243 Win, .260 Rem, .30-30, .44 Rem Mag, or .357 Magnum would all be snatched up by me. These came with a tapered octagon barrel, and weigh only about 6.25 lbs. The .30-30, .44 and .357 came with open sights, a flip up tang peep sight for longer range work, oil finish wood, and crescent steel butt, with case coloring.