The baby prince, who was born on Tuesday evening, will become Duke of Södermanland, the region south of Stockholm.
“It is great that the stork has been active for the Bernadotte family this spring,” the King told Swedish journalists gathered at the Royal Palace, referring to the birth of Prince Oscar just seven weeks ago.
Prime Minister Stefan Löfven, who attended the press conference, revealed he had met the little prince earlier that day and said he had held him for 15 minutes, describing the experience as “wonderful”.
“He was asleep, but had apparently kept his parents up during the night,” said Löfven.
Princess Sofia, Prince Carl Philip and Prince Alexander. Photo: Kungahuset/Royal Court
On Wednesday the first picture of Prince Alexander, as he will be referred to, and his parents Prince Carl Philip and Princess Sofia leaving Danderyd Hospital was released. The happy family is currently staying at Drottningholm castle west of Stockholm.
His grandmother Queen Silvia was in New York during the birth, but the King paid a visit to the hospital on Wednesday morning to say hello to the new prince.
“What's important is that Princess Sofia and Prince Carl Philip are well. This little child too. Looked very nice and cute,” the King was quoted by the Expressen tabloid as saying while leaving the hospital.
“For my wife and me, this is a great day full of emotion,” Carl Philip told a press conference following the birth of the couple's first child late on Tuesday.
The baby weighed in at 3.6 kilograms (8 pounds) and measured 49 centimetres (19.3 inches), the prince added.
Carl Philip, 36, is the second child and only son of King Carl XVI Gustaf and Queen Silvia of Sweden. His elder sister Princess Victoria is first in line to the throne.
There had been rumours ahead of Thursday's press conference that their son would be named Duke of Dalarna, the region where Sofia grew up.
Alexander already has plenty of royal cousins to play with. Prince Oscar, the second child of Princess Victoria, was born just seven weeks ago. He is the younger brother of four-year-old Princess Estelle. London-based Princess Madeleine also has two children, Princess Leonore and Prince Nicolas.
A total of 77,817 men and 29 women are called Alexander in Sweden.