I have been in this situation twice (people must really REALLY want me
NOT). Seriously, I think it happens because some companies just don't make proper plans.
On the first occasion I explained that I had to give adequate notice to my current employers, that since they had offered the job to me they were eager to appoint me as opposed to someone else and that they, I was sure, would like an employee to give them proper notice. I put it all a little more tactfully than that and managed to give the proper notice to my then employers.
The second time this happened was when I was being appointed to my current post. I managed to work three weeks out of the four weeks notice I had to give and my employers were really understanding. I explained that the new post was a great opportunity for me. It was especially kind of them because I had worked there for nine years and I knew they would find it hard to cover my post so quickly, even with a temp.
Raindance