The Early Gold Wars

The Congress shall have power ...to coin Money, regulate the Value thereof, and of foreign Coin, and fix the Standard of Weights and Measures." 
1787 - The Constitution of the United States - Section 8

That is precisely what the Congress did. In 1792, the Dollar was fixed by law at 24.75 grains or 0.05156 troy oz. of Gold. In 1837, the coinage was reworked and the Dollar was defined at 25.8 grains of Gold "nine-tenths fine". That gives 20.67 Dollars to one troy oz. of Gold. That was the Dollar's "fixed value" (see the quote above) for 96 years from 1837 to 1933.