Her Majesty Queen Margrethe II has on her bithday (16 April) decided to give Princess Alexandra an additional title. She is now: Her Highness Princess Alexandra, Princess of Denmark, Countess of Fredriksborg. Or in short, H.H. Princess Alexandra, Countess of Fredriksborg.
The press release clarifies that the divorce between Prince Joachim and Princess Alexandra does not change her status as a member of the Royal House, and that she will remain a Princess of Denmark with the prefix Her Highness until the day of an eventual re-marriage for her part should come.
The title Countess of Fredriksborg is Alexandra's personal title (a personal title cannot be passed down I think), and if she should re-marry she will keep this title and be ranked as Class 1.1. which would make her styled Her Excellency.