Posts page duplicate headings instead of blog page name

Hi, I’m using the method detailed here:

http://codex.wordpress.org/Creating_a_Static_Front_Page

… so that I can have a “static” front page and a separate blog posts page. I’ve got it working, except that the posts page ends up with the latest post’s title as the h1 rather than (say) “Blog.”

This is despite the fact that I have attempted an alteration of the default index.php, which I have called blogindex.php and placed the following code in the head:

<?php /* Template Name: Blog Index Template */ ?>

In the main part of the blogindex.php page, I killed the usual PHP call for an h1 in favour of manually typing the blog’s name. Despite that, this gets ignored and the page still displays the latest post’s heading as the h1.

What do I need to do? I don’t want the latest post’s title as my h1….