Any COPY of the old database being used to populate the NEW database will have (by default) a table prefix, if one was used, in the CREATE TABLE sql query.

EG:

Code:
CREATE TABLE IF NOT EXISTS `zen_address_book` (
  `address_book_id` int(11) NOT NULL AUTO_INCREMENT,
  `customers_id` int(11) NOT NULL DEFAULT '0',
  `entry_gender` char(1) COLLATE latin1_general_ci NOT NULL DEFAULT '',
  `entry_company` varchar(64) COLLATE latin1_general_ci DEFAULT NULL,
  `entry_firstname` varchar(32) COLLATE latin1_general_ci NOT NULL DEFAULT '',
  `entry_lastname` varchar(32) COLLATE latin1_general_ci NOT NULL DEFAULT '',
  `entry_street_address` varchar(64) COLLATE latin1_general_ci NOT NULL DEFAULT '',
  `entry_suburb` varchar(32) COLLATE latin1_general_ci DEFAULT NULL,
  `entry_postcode` varchar(10) COLLATE latin1_general_ci NOT NULL DEFAULT '',
  `entry_city` varchar(32) COLLATE latin1_general_ci NOT NULL DEFAULT '',
  `entry_state` varchar(32) COLLATE latin1_general_ci DEFAULT NULL,
  `entry_country_id` int(11) NOT NULL DEFAULT '0',
  `entry_zone_id` int(11) NOT NULL DEFAULT '0',
  PRIMARY KEY (`address_book_id`),
  KEY `idx_address_book_customers_id_zen` (`customers_id`)
) ENGINE=MyISAM  DEFAULT CHARSET=latin1 COLLATE=latin1_general_ci AUTO_INCREMENT=3 ;
If you wanted to set up the NEW database without the zen_prefix, you would either have to manually remove the zen_ prefix from the CREATE TABLE line:

Code:
CREATE TABLE IF NOT EXISTS `zen_address_book`
becomes...

Code:
CREATE TABLE IF NOT EXISTS `address_book`
or... once the database is created, run a query that removes the prefix from ALL table names.

You must then ensure that the references in the two configure.php files (for the new shop) make no reference to a prefix:

Code:
  define('DB_PREFIX', 'zen_');
becomes:
Code:
  define('DB_PREFIX', '');
As long as it all matches up, it should work fine.