Getting child menu items based on $mid (Drupal 6.x) ?

This used to be possible simply in 5.x, however in 6.x the API has changed considerably (retrieval by name for menu items?)

What I'm trying to do

I have a primary links menu consisting of 6 links. On a certain page of my theme in a certain position, I'm trying to render out the sub-tree of just one of these primary menu links.

I know the menu id for this item (but the name can change), but as of 6.x, not how to render out the child items of this menu-item. Does anyone know how to do this? I haven't found a usable solution.

Ideally I'm after something like (psuedo-ish-code follows)

//Build tree. Mid is 231
$submenu_tree = menu_tree($mid); 

//Send to my theme function for much win
print theme('links', $submenu_tree);