I can't find any sort of currency act for the Americans (not like the many wikipedia sites are helping either, getting in the way of the search results). But all I could find were:
No State shall enter into any Treaty, Alliance, or Confederation; grant Letters of Marque and Reprisal; coin Money; emit Bills of Credit; make any Thing but gold and silver Coin a Tender in Payment of Debts; pass any Bill of Attainder, ex post facto Law, or Law impairing the Obligation of Contracts, or grant any Title of Nobility.
These clauses permit Congress to coin money and to issue paper currency. By extension, under its ability to enact laws "necessary and proper" to carry out these powers (as stated at the end of Article 1, Section 8), Congress created the Federal Reserve System to regulate the nation's monetary supply.