This seems to work. :)

use strict;
use warnings;
use Data::Dumper;
use URI::Find;

my $data = do {local $/;<DATA>};

my $finder = URI::Find->new(
    sub {
        my($uri, $orig_uri) = @_;
        return qq|<a href="$uri">$orig_uri</a>|;
    }
);

$finder->find( \$data );

print $data;


__DATA__
Blah blah blah
Web: http://www.example.com
info.example.com
Web: http://info.example.com/
And another
Web site: http://another.example.com/ (doesn't always work)
Nice blog here: http://blog.example.com/niceblog/
[download]

UPDATE: change the inner sub to this to get exactly what you want:

sub {
    my $u = URI->new( $_[0] );
    return sprintf( '<a href="%s">%s%s</a>', $u, $u->host, $u->path );
}
[download]

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)