Your reply was helpful. I now have the following to create a singleton database handle.

package MyProject::DbhSingleton;
use Apache::Singleton::Request;
use base (qw Apache::Singleton::Request);
sub _new_instance {
  my $dbh = DBI->connect($dsn, $user, $pass, $attr) or die DBI->errstr
+;
}
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This would work if I called it within every method (sub) that does database access, but I'm looking for a way to avoid going to that extreme.