Or alternately you could have your form on nodatabase.com call a script on databasehere.com as its action - and then set a header("location: ...") redirect to send it back to the page of your chosing. I just tested it and the databasehere.com url never shows up.
In nodatabase.com/form.php
<form action="http://www.databasehere.com/email.php" method="post">
...
In databasehere.com/email.php
/*
* Code for processing information
*/
header("Location: http://www.nodatabase.com/return.php");