* make a backup of a mysql database called DBNAME with:
 mysqldump -uUSERID -hDBHOST -pPASSWD --opt DBNAME > DBNAME.mysql

 * create a new database with:
 mysql> create database NEWDB;

(user USERID must have CREATE permissions) 

 * unpack the backed-up copy of DBNAME into your new database NEWDB with:

 mysql -uUSERID -hDBHOST -pPASSWD NEWDB < DBNAME.mysql
(will wipe out the exising DB)