I would like info on this too. I LOVE begonias but I have the same problem where they flower stalk totally rots away. I don't think its from overwatering though. How can I prevent this.

I'm afraid it is bad news, I've never had anything recover from soft, brown looking parts. What you might do is cut a piece above the rot if one still looks firm and normal, and propagate it. They can rot so easily,you may not recognizeit starting.If they are well watered but you have some unusual cold or rainy days, you should watch for problems. It is better to lift/or touch the soil to see if the plant really needs water, if it looks okay wait another day to water, it won't collapse overnight.Believe me, I've lost my share. There is a chance it may grow again from the soil level or just below, depends on how bad the rot is and there's no way to tell, just watch and see (cross your fingers).

I just brought a begonia back from the dead, after killing several of them in the past (through over watering)! I wrote about it in a blog post here, I was so excited!
http://plantkiller.wordpress.com/2011/08/01/it-dont-begone-yet/

Basically, I had two waterlogged wax begonias and I let them completely dry out and replanted one outside in a sunnier area (it was in a pot in a shadier spot). It's blooming now!

Good luck, I had a whole bunch of begonias rot out and die on me last year.

Here is a link that might be useful: plantkiller