Amazon Prime is a membership program that offers unlimited free two-day shipping on certain products for $79 a year. Amazon set a record on orders globally, as well, with more than 36.8 million items ordered on Cyber Monday, or 426 items per second, alone.

Amazon said more than 1 million customers became new Amazon Prime subscribers in the third week of December.

Amazon.com on Thursday said it limited new Prime membership signups during peak periods in order to protect current member orders from the surge in new memberships.

The news comes as shipping giants United Parcel Service and FedEx struggled—and in some cases failed—to get packages delivered before Christmas. The delays were blamed on poor weather earlier this week, as well as overloaded systems. The holiday shopping period this year was shorter than usual, more buying was done online and Americans' tendency to wait until the last possible second to shop probably didn't help either.

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Neither company said how many packages were delayed, but noted it was a small share of overall holiday shipments.

Responding to the delivery delays, Amazon issued a statement Thursday: "Amazon fulfillment centers processed and tendered customer orders to delivery carriers on time for holiday delivery. We are reviewing the performance of the delivery carriers." The retailer also said it will refund any shipping charges associated with impacted shipments, and offer a $20 gift card to customers who did not receive packages on time.

CORRECTION: An earlier version of this story was corrected to reflect that Amazon limited new Prime membership signups during peak periods in order to protect current member orders from the surge in new memberships. It was also clarified to show that Amazon set a record on orders globally.

—By CNBC. The Associated Press contributed to this report.