A Falling Block Works Model K, with a light barrel and reasonable wood on it will come in at around 5 pounds. Less with a pencil barrel. Guys have built them as heavy as a 7-30 Waters and similar cartridges.
A Fraser Highlander falling block will do the same. Figure 5.5 with a scope on it, again with the pencil diameter barrel. Or less.
These are both falling block single shots sized towards a Hornet or Bee cartridge. The "K" looks like a mini Win Highwall, while the Highlander looks like a shrunk down Ruger #1, sorta.
Not gonna get a 300 Win Mag in either action, though. More on the level of 17 or 22 Hornets, or a .256 Winchester, or .218 Bee kinda cartridges.
A Stevens Favorite in .32 Short would be a 4 1/4 pound centerfire rifle.

A TC Contender G2 is a bit more than a pound heavier.
One of the H&R Handi Rifles in a Lightweight model will come in at about 5 1/2 pounds.
A Martini cadet action can be built as light as this, too, and can handle modern High pressure cartridges based on the .223. It'd be another custom build...
What kind of centerfire rifle, to do what, with?
Cheers
Trev