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So when will you know its safe to delete the old one ??.There is an easy way to find. Just change the content of a non important page like "About us" in the new server a little (like putting one are two extra words ). So when after the Google index update if you see this changes in the index its time to say good bye to the old webhost . But if you want to be 100% safe wait for another deep crawl .
I timed the move by waiting for the deep crawl to appear in the logs and then requesting the move.
Six weeks later the logs have shown a deep crawl and an update retaining the good positions. I believe Google looks up DNS about every 2 or 3 crawls, so there may have been an alement of luck involved.
The client was warned that a dip out of Google for a couple of months may be the price of rescuing the site from a very poor and limited ISP. However it was all seamless.