The 43-year-old, capped 67 times by his country, takes over from Brian McDermott, whose second spell in charge of the club lasted just six months.
A coach at Ajax last season, Stam had been favourite to succeed McDermott.
"Reading are very ambitious like I am and want to get back to the Premier League," he said. "Hopefully we can have a good partnership and get there."
Stam, who was part of United's 1998-99 treble-winning side during his three years at Old Trafford, joins the Championship club on a two-year contract.
The former Lazio, AC Milan and Ajax centre-back's new role with the Royals is his first in management.
"Everyone knows that teams in the Championship are very competitive," he added. "It's not going to be easy, but we'll work very hard."
He becomes Reading's first non-British or Irish manager. Dutch duo Andries Ulderink and Said Bakkati will join his coaching team having worked with him at Ajax.
First-team coaches Steven Reid and Dave Beasant will both remain in their positions.