Sweden's youngest royal also has been granted the title Duke of Södermanland, the royal palace has confirmed.
"It's an unexpected choice," royal nomenclature expert Herman Lindqvist told the Swedish outlet SVT. "No Swedish royals have been called Alexander, although during the 1800s, some princes had it as one of their names."
Prince Alexander's grandfather King Carl XVI Gustaf announced the newborn prince's name at a specially convened cabinet meeting on Thursday.
His birth was celebrated with a 21-gun salute from Skeppsholmen in Stockholm.
Family members and selected dignitaries close to the ruling house of Bernadotte are expected to attend a traditional thanksgiving service on Friday in the Royal Palace's Chapel to celebrate the birth of the new prince.
Prince Alexander, who is fifth in line to the nation's throne, was born at 6:25 p.m. on April 19.
He and his parents have already left Danderyd Hospital to return to their new home in the grounds of Stockholm's Royal Palace.
Proud dad Prince Carl Philip announced the arrival of his first child on at a press conference following the birth.
The prince, who cut the baby's umbilical chord himself, also revealed that his son was dark-haired.
"For my wife and me, this is a great day full of emotion," he told reporters.
Prince's Alexander's grandmother Queen Silvia, who will be leaving a UN conference in New York to return to Stockholm on Thursday, has already enjoyed a brief visit with her youngest grandchild via FaceTime, according to a report in the Swedish journal Svenskdam.
The latest addition to the Swedish royal family has plenty of cousins, so he will have no shortage of playmates around the palace.
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Prince Oscar, the second child of Crown Princess Victoria, was born on March 1. He is the younger brother of 4-year-old Princess Estelle.
Prince Carl Philip's London-based sister Princess Madeleine also has two children, Princess Leonore and Prince Nicolas, who joined mom in the congratulations.
"Leonore and Nicolas are thrilled to have a third cousin and we can't wait to see him!" she “wrote.