Go to the Child Site. Select the Site Actions Button->Site Settings.
You will see a menu entry for 'Save Site as Template'. That puts an
.stp file in the Site template  gallery. Download that and either use
STSADM to install it globally or if you just want to create  Child Sites
in that Site Collection then upload it to the Site Template gallery in
the Top Level Site of the Site Collection.

For some reason I do not have that menu option. I am setup as an admin on the
server. Are there special security settings that I need to have to see that
option?

Sites built off a template  that implements the Publishing feature can't be saved
as a template. These are the only ones that I have seen that don't include the
Save site  as Template menu entry. What Site Template did you use to build the
site, and did you turn on the Publishing Feature after building the site.

That is exactly my problem. I build the site  using the publishing
site template.

http://imagine-windowslive.com/hotmail/?locale=en-us&ocid=TXT_TAGHM_migration_HM\
WL_mini_pcmag_0707

The publishing site  template actually uses several site definitions to build the
site. Therefore you can't save it as a single .stp file. So the menu entry has
been removed.