OK, the line is the top of a border that your borrowed code puts around h1 a, the area to which the link applies.
Omitting
border:1px solid #FFFFFF;
should get rid of the line, but the link will still only apply to the section of the image beneath where the line is now.
To get it working properly, I think you'll need to match the dimensions of h1 a to those of #header, and remove the padding you've applied to h1 (which pushes h1 a down the page.
Adapting sunburntkamel's code, you get:
h1 a {<br />
display:block;<br />
align:center;<br />
height:200px;<br />
width:760px;<br />
text-indent: -9000px;<br />
}
Then go to h1, and delete:
padding-top:75px;
hope this helps