No! Let us assure you it's not necessary to move your web site In fact, we're not a web host, and do not even host most of our own web sites (we contract that to a web host provider, like most companies do).

You can keep your web hosting where it is, and move your e-mail by itself. This is a fundamental part of the domain names system, the ability to identify different servers on the network (internet) to take care of different jobs.

In this case, www.yourcompany.com (insert your domain name there) will take visitors to your current web host

The MX, or mail exchange record, of your domain name, will direct people to send e-mail to 01.com's Zimbra servers, where the mail will be routed into your mailbox(es). 01.com MX server settings.

All of this happens, of course, transparently to the users.  We will help you make the domain name (DNS) change once you have service established with us.  The process of switching your MX record does not require any down-time, nor will you lose mail during the process, assuming that your old mail provider account remains active during the transition.