There's no way to add a 2003 domain controller to a NT4 domain; your only option is to perform an in-place upgrade of a NT4 PDC: this will convert the domain to Active Directory, and then you will be able to add new 2003 DCs.
Converting the domain will require you to setup a proper DNS infrastructure, which is critical to AD operation; this can be automatically done during the NT -> 2003 upgrade. This will also give your domain two names instead of one: the NetBIOS name (i.e. the NT4-style domain name) will remain unchanged, but a new DNS name will be created, usually by appending some suffix to the NT4 domain name. So, if your domain is called "MYDOMAIN", you will be asked during the upgrade to supply a DNS name, which could be anything you want, provided it's not a single-label name; "mydomain.local" or "mydomain.com" are both fine choices, but usually you don't want this to be the same as your public domain name (if you have one), due to potential DNS issues.
After the upgrade is performed, you'll end up with a working Active Directory domain, but your main 2003 domain controller will be a server that has been upgraded from NT; it's usually not advised to keep it running, as it may not be completely stable and will probably not have very recent hardware. You will need to add other 2003 DCs to the domain (this time using DCPROMO), move the DNS service and the FSMO roles to them, and demote (using DCPROMO again) this server in order to properly remove it from Active Directory; you'll also need to remove NT4 BDCs, if you have them.
As you can see, the process is not overly complex, but it requires a lot of care and attention to details; also, it will require some thorough planning and testing. But it's still strongly suggested to act this way, because migrating your domain to a new one using the Active Directoy Migration Tool is a lot more complex and error-prone.
For more details, refer to the Technet articles other people have linked in their answers.