I believe it is illegal to pay off a credit card with another credit card.
Interactive Brokers offers margin at very low rates, especially if you have a lot of money. I'm paying 1.6% but it gets down to .5% if you are the billionaire type.
It's not illegal to pay off a credit card with another credit card. It's called a "balance transfer" and lots of companies will allow you to do this. The catch is that there's almost always either a fee to perform this transaction (2-4% of the amount transferred is common) or the transaction will start accruing interest immediately. You can also take out money with a cash advance one one credit card to pay off your debt on the other credit card, but that often has an even higher interest rate than purchases.
Sometimes you will see a rare credit card offer for a balance transfer with a 0% fee and 0% interest for some period of time. Lots of people do perform arbitrage with these offers, putting the money in a savings account or some other investment until the promotional rate expires. These 0% balance transfer offers were much more common a few years ago; they have all but vanished now.
Just expanding further on what was said above:
You can still find 0% offerings, but they are fewer and farbetween compared to the happy credit years (y'know before that big credit crisis thingie)
It's not illegal, as was said, but they get you in the transfer fees which is usually a percentage of the transferred amount (2% in a recent CC I got, minimum $10)
Considering you'd need to transfer every month to keep these two accounts afloat, that of course works out to a terrible 10-12% annual cost on the borrowed amount.
Alternatively, you could put it all on a 1 year 0% card, and then pray that your investments are up in the green by the time the year rolls around. Again, this is not your money, and you will put yourself in to deep water if you mis-step or fall behind on a minimum payment, or your investment fails to work out by the year's end and you can't cover it (It goes up to the fantastic rate of 16-25% interest when the promotional rate ends, or if you miss a minimum payment :) Yay!)
Also, credit transfers I believe are noted and somehow adjusted in to your credit score, which could impact your clearance for future CC deals / transfers / mortgages. So at some point, the bank can just deny you additional services if they note that you're floating a lot of debt on promotions without paying it off.