Agree with strengthening generally - what worked for me to get rid of this problem was specifically strengthening my glutes - leg presses in the gym, squats, lunges etc. Other bits of my legs used to occasionally fail too, eg calf problems & thigh splints, but now generally as all parts of my legs are much stronger than before, I can run as fast as I want without injury and still be able to walk properly the next day, no more walking down the stairs sideways!
Have a look at my previous posts on this, there's more detail in there.
And, purely by chance, I was reading March's Runner's World today and it said 'weak or inactive glutes can cause you to overuse your calf muscles to propel your body, which can strain your achilles tendons'
So, if and when you do go for a run next, activate your glutes first by doing some glute exercises and get them firing before heading out, that will probably help as well.
Also, if you have been out of sport for a while, your muscles will probably be de-conditioned.
I had achilles problems for a long time, years, and the thing that finally solved it was strengthening my legs, my glutes in particular - leg presses, single leg straight leg deadlifts, single leg squats and dips etc, some done in the gym, and some not. My physio identified my glutes as a weakness in my kinetic chain (I think that was the terminology), and now, since my glutes are way stronger, as are the rest of my legs, I have no achilles problems at all, or in fact any other leg injuries (no pulls etc).
As T Rex says above, your body has designated your achilles as it weakest place - stronger, more conditioned glutes will act as shock absorbers and help to take some of the load off the achilles.
I'm knocking on 50 though - you might be a bit younger - still worth a bit of leg training though, that might help at least.
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.