Overflow non da l'effetto che vuoi tu questo è il codice giusto:
codice:
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">

<html xmlns="http://www.w3.org/1999/xhtml">

   <head>
      <style type="text/css">
         #box_esterno{
         width: 640px;
         margin: 10px auto;
         border: dashed 2px #ff6600;
	   clear: both;
         }
         #box1{
         width: 300px;
         float: left;
         }
         #box2{
         width: 300px;
         float: left;
         }
	   #clear {
		   clear: both;
	   }
         body{
         font-family: verdana, arial, sans-serif;
         }
         h1{
         font-size: 30px;
         color: #ff0000;
         text-align: center;
         }
         h2{
         font-size: 28px;
         color: #ff0000;
         text-align: center;
         }
         h3{
         font-size: 22px;
         }
         p{
         font-size: 18px;
         }
         a{
         color: #3300ff;
         }
         ul{
         font-size: 18px;
         }
      </style>
   </head>
   <body>
   <div id="box_esterno">
         <div id="box1">
         <h1>TEST</h1>
         </div>
     <div id="box2">
         <h1>TEST</h1>
      </div>
     <div id="clear"></div>
      </div>
   </body>

</html>