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Princess Charlene gave birth on 10 December
The prince has been a constant presence at the hospital since his children's birth last Wednesday. However, he took a few hours off on Friday to attend an official ceremony to mark the signing of the birth certificates.
Around 20 guests attended including his sister Princess Stephanie and her daughters Pauline and Camille, who clinked champagne glasses with the proud father.
On Sunday night, the leader of the principality was cheered as he arrived to watch football club AS Monaco in action.
The Princely Palace has been festooned with flags in the national colour. Meanwhile the births were also celebrated with 42 cannon shots – 21 for each child – and the ringing of church bells.
On 7 January, the couple will show off their children from the palace balcony just as Albert's own parents Princess Grace and Prince Rainier did 56 years ago. The little girl was born first at 5.04pm and her brother following two minutes later at 5.06pm.
According to Monegasque law, which favours males Jacques is the heir to the throne.
Ahead of the birth Charlene, 36, described motherhood as "an adventure".
"Any mother will tell you of the intense feelings, the incredible emotion. I am very well and we're obviously very happy, I would say filled with happiness. The prospect of the birth is so magical, so powerful.
"We are full of love, gratitude, as well as a great sense of responsibility.
"I am particularly grateful for the spontaneous love bestowed on me in these special moments – the show of affection for my family, of the totality of the Monegasque people and all our friends around the world."