How to hide the "submit" button once a form is sent successfully?

How to hide the "submit" button once a form is sent successfully?

I'd like to hide the "submit" button of my form when the form is sent
successfully. I tried to add $('#submit').hide(2000); just after
$('.success').fadeIn(1000); but without success. Thanks
HTML:
<div class="block-right"> <h1>Formulaire de contact</h1>
<!-- CONTACT FORM-->
<div class="contact-form">
<form id="contactfrm" method="post" class="clearfix">
<div class="clearfix">
<input id="name" name="name" placeholder="Name"
type="text" value="">
<input id="email" name="email" placeholder="Email"
type="email" value="">
</div>
<textarea id="message" name="message"
placeholder="Message"></textarea>
<input type="submit" value="Envoyer" name="submit">
<p class="success" style="display:none">Your message has been sent
successfully.</p>
<p class="error" style="display:none">E-mail must be valid and message
must be longer than 100 characters.</p>
</form>
</div><!-- /.contact-form -->
</div> <!-- End DIV block-right -->
JS:
// Contact Form
$(document).ready(function(){
$("#contactfrm").submit(function(e){
e.preventDefault();
var name = $("#name").val();
var email = $("#email").val();
var message = $("#message").val();
var dataString = 'name=' + name + '&email=' + email + '&message=' +
message;
function isValidEmail(emailAddress) {
var pattern = new
RegExp(/^((([a-z]|\d|[!#\$%&'\*\+\-\/=\?\^_`{\|}~]|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])+(\.([a-z]|\d|[!#\$%&'\*\+\-\/=\?\^_`{\|}~]|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])+)*)|((\x22)((((\x20|\x09)*(\x0d\x0a))?(\x20|\x09)+)?(([\x01-\x08\x0b\x0c\x0e-\x1f\x7f]|\x21|[\x23-\x5b]|[\x5d-\x7e]|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])|(\\([\x01-\x09\x0b\x0c\x0d-\x7f]|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF]))))*(((\x20|\x09)*(\x0d\x0a))?(\x20|\x09)+)?(\x22)))@((([a-z]|\d|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])|(([a-z]|\d|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])([a-z]|\d|-|\.|_|~|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])*([a-z]|\d|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])))\.)+(([a-z]|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])|(([a-z]|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])([a-z]|\d|-|\.|_|~|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])*([a-z]|[\u00A0-\uD7FF\uF900-\uFDCF\uFDF0-\uFFEF])))\.?$/i);
return pattern.test(emailAddress);
};
if (isValidEmail(email) && (message.length > 10) && (name.length > 1)){
$.ajax({
type: "POST",
url: "sendmessage.php",
data: dataString,
success: function(){
$('.success').fadeIn(1000);
}
});
} else{
$('.error').fadeIn(1000);
}
return false;
});
});