Really, it depends how bad it is.
You may try the elliptical and see if it bothers it or not (or a stationary bike, as well). If you can just feel it coming on now for the first time, chances are it isn't bad enough that those activities will hinder the healing much.
I agree with previous posts - I like the ice bath, and the ice massage is also highly recommended. Just be sure not to overdo it and damage the skin. If the skin gets somewhat damaged, you'll have to lay off the cold therapy for a while, and that also means stopping the Achilles therapy.
The very best therapy, in general, seems to be eccentric calf raises to strengthen the tendon and related muscles. These should be performed seated and standing, since each exercise works the legs in different ways. Let me know if you need more info on those. These have been scientifically demonstrated to be highly effective, and I have personally seen good results with these through many experiences.
I would say try to do 2 weeks of rest, if you can. It is better to have 2 weeks of rest and be done with it, than have a week of rest, have it almost go away, then miss a day or two of training here or there, and then maybe another week, etc. wind up with a persistent injury. I would also recommend easing back into running - don't do 30 minutes back on your first day, just go for 5-10 minutes and make sure things feel okay. Take a few days to get back to the duration of your normal training run and make sure it does not hurt after that... if it does, back off. I know this sounds very conservative, but I really think it is worth it in the long term.
The other thing is to find out the true cause of the injury. Try to determine if you have a strength imbalance, flexibility imbalance, etc. (If you need more info on that, let me know.) Many people get caught up thinking about the Achilles itself, but there may be a problem elsewhere in your legs/feet causing you to slightly change your mechanics and place greater stress on your Achilles and injure it.