Here's a thread on another forum that addresses this question:
How to embed blog in existing web page? [Forums - Blog Newbies] : Blogger Forum
The basic answer is to use an iframe (sample code provided through the above link) on your web site, and to hack out everything in your Blogger blog that you don't want to show.
Note that this isn't limited to Blogger blogs, it's just that those are free and easy to setup, so should work well for you. You could do something similar with any free blog host that allows you to edit what shows on the screen.
Oh, and the reason it asked you to login again is that this forum has a relatively low timeout on logins, so if you take too long to compose a post it automatically logs you out.
Jay