The pop star gave birth at 9.36 p.m. to a boy weighing almost 3 kg and both the mother and child are "in perfect health", the communique stated.
Shakira was admitted to the private Teknon clinic at 6.30 p.m. accompanied by her boyfriend, F.C. Barcelona soccer star Gerard Pique.
Shakira shows off her pregnant belly with footballer boyfriend Gerard Pique. Picture courtesy Shakira's twitter account
In the communique, the couple and the family said: "We are happy to announce the birth of Milan Pique Mebarak, son of Shakira Mebarak and Gerard Pique ... The name Milan (pronounced MEE-lahn), means dear, loving and gracious in Slavic; in ancient Roman, eager and laborious; and in Sanskrit, unification".
Since early Tuesday morning, numerous reporters and photographers had remained on alert outside the clinic, given recent rumours that this would be the day the child would be born.
On Tuesday morning, before the birth, Shakira had tweeted, "I'd like to ask you all to accompany me in your prayers on this very important day of my life."
The baby was delivered by previously-scheduled Caesarean by a medical team headed by Dr. Carlota Garcia-Valdecasa.
Shakira, 35, last appeared in public Jan 14 at the Casa del Libro in Barcelona, where she attended the presentation of a book written by her father, William Mebarak.