Yes Apple will send you a new phone and you send them the old one back. It's called their Advance Replacement Service. If you have AppleCare on the device, the service is free. If you don't, I think it's a $30 charge? Likely to cover shipping. They will put a hold on your credit card for the full amount of a new phone, but the hold comes off when they get your old one back.
I live 5 hours from an Apple Store so I have only used this service and have never taken a phone in to a genius. I've called AppleCare as late as 6pm and after them determining I needed a replacement, I still had a phone at my door the next day. They give you a box and you pack up your old one and call UPS/FedEx (or whatever courier they use in your country) and they pick it up. Or you can drop it off for the courier.
It has taken as long as 2 days for me to get a new phone. And by new, I mean refurbished (I would assume). Have only had a problem with one refurb for my daughter that would constantly reboot. They replaced it immediately.