I resigned and my employer continued to pay me after I left the company. I received a registered letter notifiying me of the overpayment and the amount that I owe the company. The amount is approximately a paycheck and a half. Do I have to pay all of this back all at once? How long do I have to pay it back?

I talked to someone in the HR department (who didn't seem to know what she was talking about) that indicated that I really didn't have to pay it back (?!?) and also said that it was a tax advantage to have overpayment.

Any advice appreciated!

Legally, yes you have to pay it back, and you have to pay it back at whatever time they tell you. The law does not give you the right to keep the money or to make payments over time.

If they choose to allow you to keep it, that is up to them. I would strongly suggest that if they tell you, you don't have to pay it back, that you get it in writing because they could come back to you later and claim they never said so. Since you don't have a legal right to the overpayment, you'd be up a creek.