I agree, changing shoes can help with achilles problems - if someone already has a pre-existing achilles problem, using the wrong shoes could make the problem worse or at least prevent the problem from getting better. It's always worth trying a different shoes just to see if it helps in any way.
In my experience, a combination of strengthening my legs and using a lighter, less structured shoe, with just a touch of support and a different heel-to-toe drop, means I now land more mid-foot than heel, and feel like I'm spending less time in contact with the ground, which can only be a good thing.
You're probably looking at at least a month's rest, I would say, probably more, based on my experience - if it's the first time you've done it, I'm guessing it may heal quicker than a re-occurrence of a previous condition. The key thing is, when you do run again, if it still hurts, and particularly if it's stiffened up the next morning, then it's time to stop again and give it a longer rest.
I've just done two runs again, after resting my achilles for nearly two months and doing lots of gym work, and I was completely fine the next morning both times, so I will continue to run, but not go overboard. Previously, I would run, then be a bit stiff the next morning, leave it a day or two, run again, and then be stiff again. I know not do do that any more.
Probably best not to stretch it for a short while, give it a bit of a rest and then gradually do some light stretching.
Mercedes has mentioned above about changing running style and landing mid-foot, which I agree will help with achilles problems - strengthening the upper legs, in my experience, will help with driving forward more efficiently as you run, particularly as you pick the pace up - my knees seem to come up higher now and it seems more natural to automatically land mid-foot when my feet come back down (I noticed this effect a while back when I was just doing walking lunges, the extra gym work on the upper legs has helped further I think).
Also, I changed from a more clunky support shoe, which I don't think I needed really, to some lighter New Balance shoes with a lower heel-to-toe drop, and that helps to land more mid-foot as well.
The conclusion this leads to is, as you get older anyway, strengthening the legs with gym work, or at least somehow with dumbbells in the house, will make it easier to run more efficiently, faster and help to prevent or reduce the severity of injuries. I have eventually got the message after years of just running and not doing anything else.
I've had achilles problems - I recently had a mild re-ocurrence, so basically almost stopped running and just went to the gym instead to strengthen all the muscles in the legs, glutes mainly (donkey kicks, leg presses, single leg dips with dumbbells, lunges with dumbbells), but quads and hamstrings also.
I've avoided aggravating the lower leg and achilles too much by minimizing any calf-specific exercises, just giving those parts of the leg as much rest as possible. My physio suggested my achilles/soleus problems may be caused by weak glutes, so I've concentrated on strengthening the glutes and given the achilles time to recover.
Also doing high intensity interval training on a bike in the gym twice a week. I went for a run on Sunday and my main problem leg appears to be almost sorted, still a bit stiff in the other achilles, but getting there. I seem to be as aerobically fit as before, due to the high intensity bike intervals.
It's the first time I've stopped running to strengthen my legs while getting over achilles problems, based on my experience I would recommend it.
Sofi - I have done the same thing as you over the last week - just concentrated on strengthening my body in the gym rather than pounding my legs on concrete. I now have a sore upper body though.
Jane - I may have had the same thing as you I think - some part of my plantar fascia in my right foot is only half as thick/wide as it should be (diagnosed when I was 17, I assume it hasn't rectified itself since then), so when I do heel drops I have to be careful not to use too much weight (i.e. dumbbells in my hands). Without weights I'm usually ok. If I overload the foot, it tends to produce some sort of strain towards the front of the foot at the base of the toes, right where I'm standing on the step/step ladder. It's worse sitting down than running, after ten minutes of running it seemed to be less of a problem. It's happened twice in the past, and takes a few months to go away, but it does go.