You need to remove the position. That should only be used if you are trying to move content without respect to its container.
The big change was the padding on the outer div. Then you just need the inner div to fill it with width: 100% and height:100%
Try this.
<style type="text/css">
#outdiv
{
margin: 20px auto 20px auto;
padding: 15px;
width: 1100px;
height: 630px;
background: #1C4675;
font-size: 14px;
-webkit-border-radius: 15px;
-moz-border-radius: 15px;
border-radius: 15px;
}
#outdiv_inner
{
width: 100%;
height: 100%;
background: #E8E8E8;
font-size: 14px;
-webkit-border-radius: 15px;
-moz-border-radius: 15px;
border-radius: 15px;
}
</style>
<div id="outdiv">
<div id="outdiv_inner"></div>
</div>