When you say you have ‘created’ a sub-domain, I assume you mean that you have added a sub-domain to the DreamHost hosting system via the panel at Manage Domains. If so, then I’m afraid that the first step you must take is to delete this sub-domain and wait for it to be fully removed from the DreamHost system.
The usual procedure to achieve what you want is to simply add a DNS A-Record to the parent domain’s DNS records, pointing the sub-domain at your home servers static IP address. However, since you do not have a static IP address, the procedure is slightly different.
The first step in the procedure is to obtain a hostname from some place like dyndns.org and to keep this hostname pointed at your dynamic IP address using one of the many Dynamic DNS clients available. You won’t be using this hostname to directly access your website, it is simply required so that you have a known destination for the DNS record you will create in the next step.
The second step requires you to add a CNAME DNS record to your parent domain. This is done via the panel at Domains -> Manage Domains, click the small DNS link under your parent domain and add a custom DNS record with the following details:
Name: The sub-domain you want to use.
Value: The hostname you obtained in step 1
If everything went well, the sub-domain should now point (indirectly) at your home server. Of-course, it may take a little while for the DNS changes to propagate.
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