It seems it is still unknown, but...
ALMOST CERTAINLY: The Doctor did not tell River his name during their wedding.
At the end of The Wedding of River Song, River comes to Amy, at her Garden:
RIVER: Okay. I'm going to tell you what I probably shouldn't. The Doctor's last secret. Don't you want to know what he whispered in my ear?
AMY: He whispered his name.
RIVER: Not his name, no.
AMY: Yes, it was. He said it was.
RIVER: Rule One?
AMY: The Doctor lies.
RIVER: So do I, all the time. Have to. Spoilers. Pretending I don't know you're my mother, pretending I don't recognise a space suit in Florida.
AMY: What did he whisper in your ear?
RIVER: Oh, that man. He's always one step ahead of everyone. Always a plan.
AMY: River, what did he tell you? River.
River didn't tell...
BUT... In the following scene, the Doctor reveals to Dorium how he escaped death.
[Charnel house]
DORIUM: Is it you? It is, isn't it. It is you, I can sense it. But how did you do it? How could you possibly have escaped?
[Calisto B space bar]
CARTER [memory]: Is there nothing else we can do?
(The Doctor leaves, then pops back again with a big grin on his face.)
DOCTOR: Actually, thinking about it
[Receptor room]
(What the Doctor whispered was -)
DOCTOR: Look into my eye.
(The Doctor waves at River from the eyeball of the Teselecta.)
The episode clearly states that the Doctor did not tell River his name during the wedding. He told her to look into his eye. Of course, we can argue that THAT assumption is also a lie, but I think that would going too far down the rabbit hole.
LIKELY: The Doctor told River his name just before his "death" in Let's Kill Hitler.
(Rory and Amy move away from the Doctor, and let River approach.)
DOCTOR: Find her. Find River Song and tell her something from me.
RIVER: Tell her what?
(The Doctor whispers in her ear.)
RIVER: Well, I'm sure she knows.
(The Doctor is dead.)
There are people who argue that telling River his name would not drive her into saving him. But, as you can see in the clip, the Doctor whispers for about 5-6 seconds. I think he could have easily said this:
Tell her that my name is [tells his name] and that I love her.
River sighs as if he said something emotional (I love her) and she smiles while answering to something obvious (That River girl surely knows you love her and most certainly knows your name).
RIVER: Well, I'm sure she knows.
Telling her that he loves her would certainly drive her into saving him. He could have also slipped his name during the whispering.
PLUS, it makes sense that he told on the moment of his "death" since he really seemed to think that he would die. And it also supports the claim of the 10th Doctor in Forest of the Dead:
DOCTOR: River, you know my name.
COMPUTER: Autodestruct in ten
DOCTOR: You whispered my name in my ear.
COMPUTER: Nine, eight, seven
DOCTOR: There's only one reason I would ever tell anyone my name. There's only one time I could.
RIVER: Hush, now.
Of course, this could mean he would only tell her if he married her. Actually, I thought that when I first watched this. But still... He could be thinking that he would only tell her on the brink of death.
Well, that was a long post. I hope I have clearly shared my ideas about this question.