This kind of technique is called corpus-based concatenative synthesis - stringing together (concatenating) little segments of sound that are pre-stored and pre-analysed (in a corpus), attempting to match an input sequence. The idea of using it with a live input is something that's being explored more and more these days. See for example Diego Constanzo's C-C-Combine, and Krotos Audio's recently released Reformer plugin. Concatenative synthesis is also a commonly used speech synthesis technique, though usually in response to eg. text input.
The problem, as user22633 pointed out, is stringing together segments in a seamless and believable way. We are so attuned to the human voice that we'd pick up on the slightest discrepancies. Besides this, building the corpus for someone's voice would be much more involved than, as you suggest, just getting a sample or two of their singing. Imagine all the possible speech segments they might articulate, in every possible pitch, not to mention glides, and differences in formant quality. Then analysis of the input would involve not only pitch tracking (a non-trivial problem in its own right) but recognition of speech segments and phonetics. Basically, your idea most likely presupposes some pretty sophisticated natural language processing techniques. That's not to say this would be impossible - especially if you put very narrow bounds on the situation, as you suggest. But without a lot of development, I'd suggest the result would be a strange robotic transformation of your voice, more than a transformation into someone else's voice. But who knows, try putting a corpus into C-C-Combine and experiment!