The French and English editions of Wort - see here and here - ran a story about the possible birthplace of Prince Félix and Princess Claire's baby. Not very surprisingly the cour grand-ducale stated: "There are no rules or laws that oblige the princesses to give birth in the country [Luxembourg]." As soon as Hereditary Grand Duke Guillaume and Hereditary Grand Duchess Stéphanie have a baby of their own, Félix and Claire's little one will be in a similar situation to another Luxembourgish princess born outside of the Grand Duchy, Princess Charlotte. The daughter of Prince Charles, brother of Grand Duke Jean, and his wife Princess Joan was born in New York City.
Curiously, the French version of the article states that the cour grand-ducale has said that "the baby will be Luxembourgish as German law does not recognise dual citizenship". As we have talked about before, Princess Claire remained a German citizen even after her wedding and while the cour is right about the difficulty of acquiring the German one as an additional citizenship or an additional citizenship to the German one, it is an entirely different story for a child born of a German parent.
A person born of a parent with German citizenship at the time of the child's birth is a German citizen, neither the place of birth nor the nationality of the other parent matter in this case. Félix and Claire's baby will have dual Luxembourgish and German citizenship from the moment of his or her birth. This multiple citizenship is not a problem for German law in case of jus sanguinis, the right to nationality based on parentage. People who are Germans on the basis of descent from a German parent do not have to apply to retain German citizenship by age 23 like people born of non-EU/Swiss foreign parents in Germany.
Whether Prince Félix and Princess Claire will 'claim' German citizenship for their baby by seeking a German passport is a totally different question. The baby will, however, be a German citizen from the moment he or she is born until he or she gives up their German nationality. If the baby is born in France like Wort speculates, he or she will also be eligible to apply for French citizenship between 13 and 16 upon request by the child's parents and if resident in France continuously since age 8, between 16 and 18 upon request by the child and if resident in France with at least 5 years' residence since age 11 or at 18, if resident in France with at least 5 years' residence since age 11.
Source: Wort, Die Bundesregierung