A likely possibility: the molecules get replaced, but it's still the same cell.
[0]: http://digitalcommons.dartmouth.edu/cgi/viewcontent.cgi?arti...
The only way to get "new" DNA is for a cell to assemble a new strand during division.
There was also a paper published last month claiming DNA synthesis following learning, but I'm sceptical:
Drop me an email (gauravvman at gmail) if you're interested in this kind of stuff! Would like to know how you came across this paper.