After submitting 'Contact' message, how do I get it to stay on page and generate a text confirmation?

I've seen this question asked on Stack Overflow before, but haven't been able to get the previous solutions to work, for my site!

I have a 'Contact' page on a website (staging.twotailsmedia.com). And right now, the Contact page works and sends an email to the right place, but it generates a 'Thank you!' message on a new, separate, unformatted page. Not what I want. I want it to stay on the Contact page, generate the 'Thank you!' confirmation on the form next to the submit button.

Here's my Form's HTML from the index.html file:

<form method="POST" action="mail.php">

   <div class="field half first">
      <label for="name">Name</label>
      <input type="text" name="name" id="name" />
   </div>

<div class="field half">
   <label for="email">Email</label>
   <input type="text" name="email" id="email" />
</div>

<div class="field">
   <label for="message">Message</label>
   <textarea name="message" id="message" rows="4"></textarea>
</div>

<ul class="actions">
   <li><input type="submit" value="Send Message" class="special" /></li>
   <li><input type="reset" value="Reset" /></li>
</ul>

</form>

Here's the PHP from the mail.php file:

<?php

        $name = $_POST['name'];

        $email = $_POST['email'];

        $message = $_POST['message'];

        $formcontent="From: $name \n Message: $message";

        $recipient = "[email protected]";

        $subject = "A message from the Two Tails MEdia website!";

        $mailheader = "From: $email \r\n";

        mail($recipient, $subject, $formcontent, $mailheader) or die("Error!");

        echo "Thank You! We'll get back to you.";
}
?>

Usually I'd just get a Contact Form plugin from Wordpress but I'm trying to do a non-WP page so I can get a deeper understanding of the web development process.