The new site replaces it, and the good parts come along with you. You don't lose what matters.
Your web address (your domain) can be moved over so it points at the new site. See the Web address section on your account page. Nothing about owning the domain changes.
Old page addresses that people bookmarked can be forwarded to the right new pages. There's a redirects tool for this, and old addresses can be migrated over automatically.
One thing to watch. Your old site's company may still bill you for hosting. Cancel that with them once the new site is live, but not before.
The support team can walk you through the whole handover if you'd like a hand.
Why this matters
Moving over the right way means customers still find you, old links still work, and you stop paying two companies at once.
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