It could be privileges issue where your new host is not allowed to access db.
If you didn't do it already try issuing command from mysql client, logged as admin:
GRANT SELECT ON mydb.* TO 'someuser'@'%' IDENTIFIED BY 'some_pass';

If you don't use linux, there are windows mysql clients too.

If they really don't allow access to external database, you can move the db files to your server or change the host.

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Impossible? You just underestimate the time.