Having Custom URL (Co-Branding, White Labelling)

CUSTOM URL: The custom url could be either the service provider giving a sub domain to the branch.(like wordpress has given me pranshujain.wordpress.com , or it could be a subdomain of the brand , or it could be a completely different domain.
Each method has different costs and considerations .

the subdomain of the service provider is definitely the cheapest. You don’t have to buy a new domain. though I am not an expert,I believe that some dns can support wildcards,redircting all requsts for all sobdomains to the same server. And if your site has secure pages,you don’t need to buy new SSL certificates, as ssl certificates do support wildcards.
At the application level you could choose to have a different instance of the web server, or you could just handle it at application level. The  negative (if it is a negative ) is that Users can’t use IP address to access the site.
I will write later about  how WCMS delivery system support this.
Another negative , rather consideration is that it can’t share cookies with brand and javascripts on either side can’t manipulate objects served from other server due to browser security .
BRAND’S SUBDOMAIN : Havlng Brand’s subdomain helps by allowing sharing of cookies. This also means that the brand has to provide ssl certificates. the service provider could again choose to have new web server, or virtual servers or read the same at the application level. the DNS has to be configured by the brand.
NEW DOMAIN: requires new DNS setting ,buying new domain and a new ssl. The modes of application sharing (or not) are same as above options.
Architects:
Consider cookie sharing and Javascript interop.
Consider a wildcard ssl or a new ssl certificate
Consider dns setting
Consider same or different application instance  -and appropriate  consideration for ui framework .

On a final note, I must also mention that a reverse proxy could be used to map a.b.com to www.b.com/a , completely transparent to the user.

Leave me a note if you need any more details. I will write about wcms support for the same later.