Exposure therapy worked for me. I was afraid of heights and had to hold someone's arm and look in at the middle of the road in order to cross the clifton suspension bridge without simply falling over, such was the giddiness.
Anyway, when I subsequently moved to Bristol I had to start crossing it by bike to get to the trails but this was ok as I was inside on the road (not the footpaths which freaked me out). I started dismounting and forcing myself to walk the paths. I also took up indoor climbing, somehow being indoors it felt less severe.
Eventually I got onto lead climbing (you put the rope in as you go), for those who don't know, the risk here is if you fall while clipping into the next carabiner, you will fall not only to the next one, but the same amount again in slack rope.
Anyway, my first big fall (I think they call it a "wanger") was a good five meter drop against the wall, it hurt a quite bit as I got proper scraped up, but ultimately, it didn't feel as bad as I'd always worried about. And after that, my thing with heights was no big deal – that was my final cure if you like.
Anyway, years have gone on and I now live the other side of the clifton bridge and have to walk across every day. I realised the other day that I can even walk along it looking straight up in the air without getting dizzy and falling over now. I'd even say I quite enjoy heights now!
That was a bit of a long boring story but hopefully CG, it will inspire you to get over yours!