fred.pl
use MyMod;
MyMod::db->fred(MYMOD_DEFAULT);

##</code><code>##

MyMod.pm
use strict;
use warnings;
package MyMod;

use constant MYMOD_DEFAULT => 1;
use Exporter qw(import);
our @EXPORT = qw (MYMOD_DEFAULT);
# changing the use to require did the trick:
require MyMod::db;
sub dave {
        print "The constant is ", $_[1], "\n";
}
1;

##</code><code>##

MyMod/db.pm
use strict;
use warnings;
package MyMod::db;
use MyMod;

sub fred {
        print "hello\n" if (MYMOD_DEFAULT);
}
1;