First, your check for the "Links" directory is a bit awkward, we shouldn't have to bail if the directory already exists:

my $dir = 'Links';
if (-d $dir) {
   chdir $dir or die "can't chdir: $!";
}
else {
   mkdir $dir or die "can't mkdir: $!";
}
[download]

open FILEIN, "urls.txt" or die "Could not open file $!";

while(<FILEIN>) {
   chomp;
   ...
}
[download]

Next, pull in the URI::Find, Config::IniHash and File::Basename CPAN modules to ease our burden. At this point, however, the skeleton of your script changes, so here is my version complete:

use strict;
use warnings;

use URI::Find;
use Config::IniHash;
use File::Basename;

use vars qw( @FOUND );
my $dir  = 'Links';
my $file = 'urls.txt';

unless (-d $dir) {
   mkdir $dir or die "can't mkdir: $!";
}

open FILEIN, $file or die "can't open $file:  $!";
my $urls = do {local $/; <FILEIN>};

my $finder = URI::Find->new(\&found);
$finder->find(\$urls);  

for (@FOUND) {
   my $hash = {
      DEFAULT => { BASEURL => $_ },
      InternetShortcut => { URL => $_, Modified => 0 },
   }; 
   my $file = basename($_,'.*');
   $file =~ s/(\.\w+)+/\.url/;
   WriteINI("$dir/\u$file", $hash);
}

sub found {
   my($uri, $orig_uri) = @_;  
   push @FOUND,$orig_uri;
}
[download]

jeffa

L-LL-L--L-LL-L--L-LL-L--
-R--R-RR-R--R-RR-R--R-RR
B--B--B--B--B--B--B--B--
H---H---H---H---H---H---
(the triplet paradiddle with high-hat)