First of all, I would re-write

    $line = <FILENAME>;
    while ($line ne "")
        {
        print "$line";
        $line = <FILENAME>;
        }
[download]

as

    while (<FILENAME>)
        {
        print ;
        }
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if the expected behavior is to print out every line of the file.

One way to tweak the above to get your desired output is as follows:

    while (<FILENAME>)
        {
        last if /<!-- Begin -->/;
        }
    while (<FILENAME>)
        {
        last if /<!-- End -->/;
        print ;
        }
[download]