Geez- no Browning guys here yet?
To convert to .308 you need a barrel in that caliber, a replacement booster with the appropriate aperture (It's smaller than the one for .30-06 and 8mm) and a set of spacers for the feed tray to position the rounds correctly.
An original .30-06 bolt might work, but might not if it's crispy and new, as the t-slot in the front of the bolt is narrower. The rims of the .308 (7.62x51 for the purists) are a bit thicker and may not slide freely down the t-slot of a .30-06 bolt. The extractors are the same. The easiest way to convert is to buy one of the Israeli parts sets that are still available. All the parts are set up for .308 and the top cover is modified to allow the use of linked .308 ammo. The original top covers don't like the combination of .308 and links.
The 1919 is very easily converted to .308 or 8mm Mauser from the original .30-06. With the price of ammo going sky-high, conversions are now available for 7.62x54R and even 7.62x39, but they involve more parts to be altered and are more expensive.
Mucho info available over at 1919A4.com.