What you need to do is set the exit hover function to be for the pop up, not the other one. Otherwise you get caught in that cycle where having the pop up appear triggers the exit, and then that triggers the enter... etc.
Here's a complete page as an example. I've changed a few things as I was working on it (more_info to an ID, for instance, and replaced animate with fadeIn and fadeOut, and am adding/removing a class instead of using jQuery's css function). Those could be changed back, of course.
I tested this and it works as expected in FF4 and Chrome.
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<title>Hover Snaddle</title>
<style type="text/css">
* { margin: 0; padding: 0; outline: none; }
body, html, div, blockquote, img, label, p, h1, h2, h3, h4, h5, h6, pre, ul, ol,
li, dl, dt, dd, form, fieldset, textarea, th, td { border: 0; margin: 0; padding: 0; outline: none; vertical-align: baseline; }
body { background: #fff; color: #000; font: normal 12px/1.5 Helvetica, Arial, sans-serif; margin: 0; padding: 0; }
#container { margin: 30px 0 0 0; }
.hovery{z-index:20!important;}
#more_info { background: #ccc; margin: 0 auto; padding: 10px 20px; position: relative; top: -38px; width: 355px; z-index: 0; display:none;}
#more_info p { color: #fff; font-size: 12px; line-height: 1.3; margin: 0; }
.learn_more_btn { background: #333; color: #fff; display: block; margin: 0 auto; padding: 10px 0; position: relative; text-align: center; text-decoration: none; width: 100px; z-index: 10;}
</style>
<script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.4.2/jquery.min.js"></script>
<script type="text/javascript">
$(document).ready(function() {
$('.more_info').css({'opacity' : 0});
$('.learn_more_btn').mouseenter(function(){
console.log("enter: "+$('#more_info').css('z-index'));
$('#more_info').addClass('hovery').fadeIn();
});
$('#more_info').mouseleave(function(){
console.log("Exit: "+$('#more_info').css('z-index'));
$('#more_info').removeClass('hovery').fadeOut();
});
});
</script>
</head>
<body>
<div id="container">
<a href="#" class="learn_more_btn">Learn More</a>
<div id="more_info">
<p>Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.</p>
</div>
</div>
</body>
</html>