Hi, the solution can be found at:

http://search.cpan.org/~capttofu/DBD-mysql-4.023/lib/DBD/mysql.pm

There you find:

DATABASE HANDLES

The DBD::mysql driver supports the following attributes of database handles (read only):
$errno = $dbh->{'mysql_errno'}; <---- SOLUTION!
$error = $dbh->{'mysql_error'};
$info = $dbh->{'mysql_hostinfo'};
$info = $dbh->{'mysql_info'};
$insertid = $dbh->{'mysql_insertid'};
$info = $dbh->{'mysql_protoinfo'};
$info = $dbh->{'mysql_serverinfo'};
$info = $dbh->{'mysql_stat'};
$threadId = $dbh->{'mysql_thread_id'};

Use $dbh->{'mysql_error'} instead of $DBI::errstr like here:

my $sql = "delete from not_existing_table where ai = 12";
$dbh->do( $sqlbef ) || die "Problem: $dbh->{'mysql_error'}\n";
[download]

you will get the answer:

Problem: Table 'not_existing_table' doesn't exist