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You can't just switch one url.
The server you have now resolves to an ip address - like example.com=18.104.22.168
your new host will have a different ip address for you.
Assuming you own the domain name - you need to get a new host and get the dns servers for their company.
This will be somthing like:
Then you tell your domain name comany (like directnic, network solutions) to change your dns settings from:
then you will have c couple of days were stuff goes to the old host and new host - as long as you leave your old stuff up for another day or two - you should get all your email.
I can tell you there is no TECHNICAL reason why you wouldn't be able to switch email from one host to another.
If this is YOUR DOMAIN. You can't switch if it is a subdomain of someone elses or something like that.
If you OWN the DOMAIN - you can switch it.
If you have control about your old DNS entries, then you can take measures to shorten the cycle. Ideally, you would set the TTL of all your records to one day about two weeks before the switch. This means that conforming DNS servers will drop their cached records about that domain and request an update every 24 hours. There will be some that ignore the TTL information and still cache the records longer than specified, though. You can give the *new* DNS entries a TTL of about a week right away, which is a typical value chosen for efficiency reasons.
Ideally, you would then keep your website up at the old location for a week or two (or redirect it to the new IP), and it's also a good idea to forward any mail still arriving there for a while.
Note that there are seperate DNS records for web access (A = "address") and e-mail (MX = "mail exchange"). You need to make sure that both are configured correctly with your new DNS service. But since in most cases the DNS services are rendered by the hoster, that shouldn't be a problem.
Once that's all done, have the current host point DNS for both WWW and Mail to the new location (you'll need the IP of your new web space).
Wait a day or so to let the routers catch up then check the new website. Verify you're looking at the new website by using tracert or ping from the command line. They will return the final IP address for the DN (www.yourdomain.com). It should match the IP of the new location. Verify your email after the change has propogated. Make sure the new host is ready to handle DNS for your Domain.
Resolve any issues with the new hosting setup now. If the problem will take few days, have the current host switch the DNS back to them.
Once the new location is working correctly, web & email, wait a few more days and then submit the modification request for your DN record so it points to the new NameServers (the NameServers of the new host). This change will take a few days to be accepted and then to propogate through the Inet. Once the mod goes through and the DN record shows the new location wait another week and then you can safely cancel your account with the old host.
The reason for changing where DNS points to is that DNS changes take only a minimal amount of time and while you're waiting the current site is still live. Once it changes the new site is already there and ready to go. If a problem arises, you switch DNS back to the old site and your downtime is kept to a minimum.
This same logic applies to when you make the DN modification. Both locations have DNS set up so that while you're waiting for the DN mod to go through, the old host is still handling DNS but as soon as the change does occur, the new host is ready to take over.