I would concentrate on a non weight bearing warm-up at least... or maybe just try some non-weight bearing exercises for a while.
Resting will help it to recover but i'd imagine you'd need to build strength into it again to prevent the injury re-occurring. Whilst probably most runners use a steady jog to warm up, putting so much strain on the achilles from cold probably will cause some pain/discomfort. Something like cycling will use the tendon and help strengthen it, without having to support the whole weight of the body.
Certainly not the definitive answer, but perhaps something to consider
