The original form of the baba was similar to the babka, a tall cylindrical yeast cake. The name means 'old woman' or 'grandmother' in the Slavic languages, and has nothing to do with Ali Baba; babka is a diminutive of the same word.
The modern version "Baba au Rhum" (Rum Baba), with dried fruit and soaking in rum, was invented in rue Montorgueil (Paris, France) in 1835 or before, both in name and in fact pretty much the original baba soaked in rum. Today, the word "Baba" in France and almost everywhere else outside eastern Europe usually refers specifically to the rum baba.
The original Baba was introduced into France in the 18th century via Alsace and Lorraine. This is attributed to Stanislas, the exiled king of Poland . However, legends crediting of Stanislas with the innovation of the "Baba au Rhum" are certainly false, theories giving partial credit to Stanislas seem possible, though partially unlikely: the Larousse Gastronomique reports that Stanislas had the idea of soaking a dried Kugelhopf (a cake roughly similar to the baba and common in Alsace-Lorraine when he arrived there) or a baba with alcoholic spirit. Another version  is that when Stanislas brought back a baba from one of his voyages it had dried up. Nicolas Stohrer, one of his pâtissiers (or possibly just apprentice pâtissiers at the time), solved the problem by addition of adding Malaga wine, saffron, dried and fresh raisin and crême pâtissière. Courchamps states in 1839 that the descendants of Stanislas served the baba with a saucière containing sweet malaga wine mixed with one sixth of Tanaisie Licquor.
Nicolas Stohrer followed Stanislas' daughter Maria Leszczyńska to Versailles as her pâtissier in 1725 when she married King Louis XV, and founded his Pâtisserie in Paris in 1730. One of his descendants allegedly had the idea of using rum in 1835. While he is believed to have done so on the fresh cakes (right out of the mold), it is a common practice today to let the baba dry a little so that it soaks up better. Later, the recipe was refined by mixing the Rum with aromatized sugar syrup.
In 1844 the Julien Brothers, Parisian pâtissiers, invented the "Savarin" which is strongly inspired by the "Baba au Rhum" but is soaked with a different alcoholic mixture and uses a circular (ring) cake mold instead of the simple round (cylindrical) form. The ring form is nowadays often associated with the Baba au Rhum as well, and the name "Savarin" is also sometimes given to the rum-soaked circular cake.
The baba was later brought to Naples by French cooks, and became a popular Neapolitan specialty, under the name babà or babbà.
The pastry has appeared on US restaurant menus since 1899, if not earlier.
It has been a while since I made the Baba, and the day has come and I made a big batch of it, and since I didn't have the right baba molds i used ramekins and then parchment papers to extend the length and give room to the dough to rise taller. I even used the popover pans and they came perfect.
the sauce was a combination of fresh orange juice, and Golden Rum.
you can dip the baba in the warm sauce and they will double again when they soak the yummy syrup. Stuffing the baba is up to you , either with pastry cream of whipped cream, and you can use a pastry bag with a fine round tip inserted in the bottom of each baba and fill it with the cream, or you can add the whipped cream on top of the baba and serve with it nuts and dried fruits soaked in the same syrup...