Yes that's a little bit confusing isn't it. The datasheet does say your input can range from 0V - VDD so if you're powering it from a +5V supply then I guess your signal can be between 0-5V D.C. The diagram you're reffering to where VCC/2 is shown is actually a diagram of the internal operation of the IC - VDD/2 must be derived directly from the supply rail inside the chip itself.
I can't say I'm 100% certain on this point, but I think you can input a voltage that spans from 0V to VDD. Personally, I'd buy one in, set it up and have a look! Just connect a POT between the supply rails, connect the potential divider pin into the ADC and have a look what happens to the digital outputs when you set certain voltages.
In any case, did you say that your sensor output is 0V-1V? If so, we'll need to do some signal conditioning anyway because you'll want to use the full window of the ADC. If you don't know what the output of your sensor is, then you'll need to find out before you can proceed.
Brian