If by 'existing' you mean 'physical', then yes, Descartes thought he could prove his existence as a 'thinking thing'. Frankly, since he would have to think to do so, it's in the bag; but it doesn't follow that he therefore exists as a physical being.
Well, Descartes' project was to discover a sound premise on which to build a logical edifice, much as Euclid had done with his axiomatic geometry. 'I think, therefore I am' was a promising start; unfortunately nothing follows from it unless you introduce other premises which don't have the same logical necessity. Descartes tried 'A good god exists', but despite throwing the ontological argument at it, he couldn't make it stick.
'I think, therefore I am' was the product of Descartes' 'method' of doubt, according to which he establishes that it is possible to doubt everything, or at least hypothesise explanations that only nutcases take seriously, but when it came to doubting, he couldn't doubt that he was doubting. So he decided that whatever was doing the doubting, was him.