Pour the yeast in a cup over the lukewarm (not hot) milk, to activate it. In a large bowl, work the yeast with 1 cup flour to form a lot of gluten. Some recipes suggest to let the dough rise a little before adding the other ingredients, but with many yeast types this is not necessary. Add the rest of the flour and the eggs, the sugar, and work well. Then add the melted butter. Knead the batter for a while - you do want to form gluten. The batter is sticky and the easiest way is to use a mixer with bread or pasta hook; but big woodden spoon (and good muscles) will do as well.
Let the batter rise in a warm place. The batter should at least double in size. Depending on the temperature of the room, it may take from 30 minutes to 2 hours, or even a longer period. Meanwhile, abundantly grease and cover with sugar a round 9 inch diameter pan with a hole (you may want to do the same even if the pan is non-stick).
Deflate the batter, by pressing on it with a spoon (it will be sticky). Pour the batter in the pan, distributing it evenly. Let the batter rise again until (almost) to the top of the pan. Then bake in an oven at 350 until the top is brown That may take 30-40 minutes, but it depends on the oven, so check the cake.
To be faster, you can skip one step and pour the batter in the greased cake pan before the first rising, being careful to distribute it more or less evenly. Then let the batter rise in the pan until it reaches the top. I still have to check whether this does anything to the texture, and if either system is better.
When baked, remove the cake from the oven, and let cool. Make small incisions or remove chunks of the crust on the top of the cake (i.e. the top of the molded cake, which will become the bottom of the unmolded cake). This is necessary to let the syrup seep through.
In a microwavable cup (such as a pyrex measuring cup) mix the cups water with the sugar, then microwave it on high for 3-4 minutes, until it bubbles (you may have to do it in two steps). Or simply boil the syrup in a pan over a fire. When the syrup boils, immediately add the rhum, and pour the boiling syrup over the cake. I simply pour the syrup over the (cooled) baba while the cake is in the mold, so it will soak better. Let it absorb the syrup for a few minutes at least, then turn it and unmold, and serve at room temperature. (Note: you may want to avoid a cake stand or the like, but rather use a dish with at least a slightly raised border - just in case some syrup seeps out over time).
Note that the syrup must be bubbling, otherwise it will not be absorbed well. The amount of rhum can be varied. Nick Malgieri suggests to use a smaller quantity of rhum. But a little zing seems appropriate to temper the sweetness. Anyway, you can vary the proportions of rhum and water, and use the quantity of syrup you like. I have not experimented with other flavors (eg rose syrup), so I don't know how it would be. Incidentally, you may want to do the pouring over a sink, so there's no problem cleaning up spills. Similarly, when you unmold the cake, you may want to do a quick check by turning the mold on a dish over the sink. The syrup should be absorbed well, but you never know - this way, again, you'd avoid a sticky mess. Another thing: a hot cake out of the oven will absorb more. Which means that if you do so, you may need to use less and less watery syrup. Anyway, the doses I have given seem to be fine for a room-temperature cake.
The cake can also be covered it with an apricot jam glaze. (Mix a couple of tbsp of apricot preserve with a tbsp sugar and a couple of tbsp water, then brush on the cake). But this is really not necessary.
The cake (without the syrup) can last a few days (covered in plastic). In fact, I usually prepare it at night a day or two in advance. It is probably best to soak the cake the day you're serving it, as otherwise the cake may slowly get dry. If kept in airtight containers in the refrigerator (airtight to avoid the cake from drying out), leftover syruped slices keep very well for several days, if not weeks, because of all the rhum and sugar. You can also freeze the cake (before soaking), as one does with bread.
The same ingredients should also produce the traditional babas. Just pour the batter in the individual molds instead of in the big cake mold, and let rise before baking. They bake faster than the big cake. To soak them in the rhum, you should drop them in a pot of boiling sugar and let them soak for a minute or so, or drop them in the microwaveable cup with the sugar, and microwave them in very bubbly syrup for 10 seconds or so. Unsoaked, they freeze well. I keep a bag in the freezer, and take one when I feel like something sweet. It takes one minute to prepare some syrup.