I have a credit card where I do this. I use it very often because it has high rewards on restaurants, so I'm constantly carrying a balance. The card is set up to automatically pay the statement balance every month. As a result, when the payment goes through, there is always a balance left (the charges that came after the statement).
I have monitored this card closely since I opened it last year to ensure that this would result in no interest charges, and it hasn't.
As Victor points out, this is due to a period of time on this card (true of many but not all cards), which is often referred to as a grace period. The "40 to 60 days" he quoted is from the time of purchase. The credit card company will give you a number of days from the close of each billing period, typically about 25 days, to pay the statement balance and avoid interest. Since your purchase could be made at any point in the billing period, the amount of time you have from purchase varies.
It's important to note that this may not apply if you are already carrying a balance that is being charged interest. If this is the case, check the terms very carefully. Often this means that interest accrues from day 1 (the day of the purchase). If in doubt contact a representative and ask specifically about your situation.
Since you mentioned that you are paying off the balance in full, you are probably not paying interest at the moment.
On a personal note, I was also paying all cards in full if they had any balance, and I found it very freeing to have the statement balance paid automatically. I still pay no interest, I don't have to worry about making manual payments, and I reduce a tiny bit of opportunity cost.