Can I pay off a balance on my Diners Club credit card if I apply for a balance transfer card from the same bank that issued my existing card?

Unfortunately, you will not be able to do this. If you have a balance with, for example, Chase Bank, you won't be able to get a different credit card from Chase and transfer that balance for a zero-interest deal. But fortunately, there are many balance transfer offers to choose from, so chances are, you'll be able to find one from a different issuing bank. Look for an offer that lasts a long time; some balance transfer cards will give you a 0% APR promotional period for a year or even longer. You should expect to pay a balance transfer fee of around 3% of the amount you transfer, but if you can find an offer that waives the balance transfer fee entirely, all the better. Just remember that if you're paying a high interest rate on your credit card balance, it will almost always be worth paying a small fee to transfer that balance to an interest-free card.