If I am understanding correctly you want people to access DomainA.com when they go to DomainB.com? If this is the case, you could set up DomainB.com as a Domain Forward to DomainA.com.

For instance a Client I have right now has www.drcharlescrane.com and www.drcharlescrane.net. Hosting is only set up for .com and we domain forward .net.

You can also have this set up as a domain forwarding with mask so if you wanted the user would actually see in the URL domainb.com but pulling domaina.com's content.

I hope this makes sense and if you need further clarification let me know. Also where is your domain registered. I use Godaddy primarily to the low costs for domains. Here is a how to domain forward provided by them and more information on the topic - http://help.godaddy.com/article/422