The Mint began taking orders for the coins again yesterday, triggering an "overwhelming" response, CNBC reports.
Sales of the American Eagles had previously been suspended last week due to the high demand, resulting in the bottleneck.
Under the new system, half of the new bullion coins produced will be allocated equally among dealers - while the remainder will be given out depending on how many each dealer has previously sold.
In a memo to dealers, the Mint said: "The unprecedented demand for American Eagle gold one-ounce bullion coins necessitates our allocating these coins among the authorized purchasers on a weekly basis until we are able to meet demand."
The bullion coins were originally introduced by the Mint in 1986.
They are popular with investors as well as collectors, as they offer an easy way of investing in the gold market.