There's probably more than one way to do this, but here's how *I* do it
BLOG A is my main blog. I have another blog called "announcements". This blog is as plain vanilla as can be. (there's not even any <html> </html> tags or <body> tags… it's just the text by itself.
Then on my main blog page, I do a php include to pull in that announcement text where I want it to show up. Assuming you have your page named with a .php extension (or your server is configured to process html files like they're php files (as it is explained here) you do a php include simply like this:
<? php
include("http://www.YOUR_SITE.COM/YOUR_SECONDARY_BLOG_FOLDER/");
?>
and that's it. Easy shmeasy.