Using base_url() on my new host.

[eluser]pickupman[/eluser]
Are you running multiple domains from the same account? I know this is typically the folder structure when you create an addon domain through cPanel. Any particular reason for creating the 2 separate applications that need index.php and admin.php. Your description of your folder layout is confusing. Try like or take a capture:
/public_html
/application
/system
/index.php
/public.php
/admin.php
/discreetdelights.co.uk
/admin.php

Maybe this would work

Code:

RewriteEngine On
RewriteBase /
##Create rules for admin.php application
    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteCond %{REQUEST_URI} ^admin.*
    RewriteRule ^(.*)$ admin.php/$1

##Create rules for public application
    RewriteRule ^(.*)$ index.php/$1 [L]
    RewriteCond %{REQUEST_URI} ^system.*
    RewriteRule ^(.*)$ /index.php/$1 [L]
    RewriteCond %{REQUEST_URI} ^application.*
    RewriteRule ^(.*)$ /index.php/$1 [L]
    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteRule ^(.*)$ index.php/$1 [L]

Or if all else fails, delete the .htaccess file, work out the file paths then add it back in. Remember .htaccess in subfolders take priority over its parent folders' .htaccess file.