time travel - When does the Doctor actually tell River his name?

People saying he said his name when they got married: remember that he was in the teselecta robot, and most certainly wasn't alone in there (remember, in let's kill hitler, the number of people there that were needed to operate it). If he said his name then, the teselecta people would know. Yeah, he could wipe their memories, as it happens in the inforarium, but why take the chance? His name is too much of a secret he needs to keep for that risk.

More likely he told her in Let's kill hitler, when he's dying and says "Find her. Find River Song and tell her something for me...". Then he whispers something in her hear, and it's either his name or that he loves River, or something like it. For what else could Mels answer be "Well, I'm sure she knows" ? Saying his name there only has one reason though: he really thought he was dying, and knew she would have to know his name for him to trust her when he first met her. (wibbly wobbly, timey wimmey...).

But, at that point, Mels was not to be trusted yet. Entrusting her with that knowledge was dangerous. She could simply babble it to the ponds, or something... She most likely didn't know yet what a secret that was, how hard it must be kept, the power of a name itself... and would not find his last words so moving as she did. Remember that after he whispers, she's quite taken aback, she finds out she's River Song and asks Amy if he is worth it, before wasting all her remaining regenerations on him. Hardly the kind of thing you would do for a person that only said their name to you... A person you were born to kill. Unless his name has indeed so much power that simply knowing it has told her everything she should know for her to change, or something like that, but that's exploring a whole different subject.

So... so much babble to say: the Doctor said his name to River Song in:

1) Their wedding : not very likely... not impossible;

2) Let's kill hitler : likely ;

3) The Last Night (or any of their nights out actually) : could be... ;

4) Some unseen situation (with the 11th) that will not be showed: likely ;

5) Some unseen situation yet to come: not impossible ;