Can I just replace my old iPhone with a new one by switching the sim card?

I broke my iPhone 5 the the other day, and was wondering if I bought a new phone directly from apple (probably a 5s) and used my simcard from the iPhone 5, will it work? And by that I mean will I have any trouble from my provider (AT&T) or from activating the phone since I'm still under contract.

Buy the phone from outside Apple and you'll save a lot of money. Just buy the phone unlocked or on AT&T and put your nano-SIM from the iPhone 5 in your new iPhone 5S and activate with your Apple ID. Simple as that. Nothing more and nothing from AT&T as you're allowed to get as many phone as you want in your contract, you're not violating anything. If you need iPhone 5S without contract cheaper than retail, mail me.