Why build one when you can build two for twice the cost? DS2 was clearly in construction while DS1 was still being built.
Most likely the Emperor foresaw the destruction of the first death star through the force. He probably even had all the traitors who poop pooped on the force and felt they were now the true power in the galaxy because of the battle station on the station so they would die with it.
This is probably also why, in part, he needed to take direct command of the second death star and put a 'yes man' in charge of its initial construction.
Or it would be very suiting for the Emperor to build two of those anyway. He was kinda megalomaniac you know...