I think what Erik means is have you tried other 8591 chips, and do they all act the same?
Your problem sounds a bit like you are getting data from another AD channel than the one you want to read.
Do you begin by writing the control byte and have the autoincrement flag set to 0 if you are interested in 1 channel only?
Remember to write a third byte with the control byte. You must put something in the DAC.
Are you using an external clock? if not: I would advise you to set the DAC enable bit.
I've used the 8591 a couple of times, and it works fine for me, I quite like this chip.
What do you use for a voltage source? A variable power supply with a DVM, or a simple potentiometer?