Nope. In it's broadest terms, your site is operating under a domain name if the initial part of any link to the site contains blahblah.com - ie words and not numbers. However, you will be sharing http://home.btconnect.com/ (for example) with millions of other site owners - which simply isn't good enough for many WordPress users. They want a unique, relevant, domain name for their site that may also give them better search engine placements.
See what your hosts offer in the way of a temporary url (in this context, "temporary" just means not a posh, unique, url). That said, now that you've forked out for hosting, you can pick up a .co.uk domain name for about £12 for 2 years - which is buttons really. Apart from anything else, it makes it a lot easier to give out a link for your site over the phone. :-)