new domain name while retaining the web hosting

I have client/customer level rights to a 3 yr. old domain + windows web hosting (IIS web server) for http://www.22shrutiharmonium.com and it has html and asp pages.

Now, I need to change the domain name to www.22shruti.com (this has been purchased from another provider), but I want to keep the files on the currrent hosting (http://www.22shrutiharmonium.com). So, only domain name to be changed, not the hosting service.

Please guide me about the step by step procedure to -

1> Carry out the domain transition.
2> While retaining (OR, minimize the decrease in) the search engine credentials of my website.

Thanks in advance.