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When you upgrade your Bitly account for the first time, you'll have the opportunity to choose a custom domain that you can use for free to shorten links in Bitly. You only get to choose one domain through this offer and cannot choose another complimentary domain after your selection is made. The domain cannot be transferred to you or another owner.
To directly own and have more control over a custom short domain, you can purchase one through a domain registrar, such as GoDaddy or Domain.com. After you obtain a domain of 32 characters or less, you can add it to Bitly for use with your shortened links.
If you're trying to transfer your own custom domain between domain registrars, please contact your registrar's support team or check their documentation. This usually involves unlocking your domain, getting an authorization code, and then transferring.