You have to pay the full statement balance that is shown on the monthly bill. If you charge more money from the time the bill was printed until the time the bill is due, you do NOT have to pay that until the next month. I repeat, you do NOT have to pay the current balance, just the statement balance.
One gotcha: some credit cards have two-cycle billing (congress was grilling them earlier this year about this horrible practice). With two-cycle billing you have to pay the current balance for two months THEN you can switch to the statement balance each month. If you don't you'll always get a little interest.