MySQL set one db in server to slave
To do this, you don't want to have any filtering on the the Master A server. Instead you will want to use replicate filters on the slave, Server B

For a full understanding of replication filters, have a read through this documentation and its sublinks. I will highlight two caveats (emphasis mine):


  On the master, you can control which databases to log changes for by using the --binlog-do-db and --binlog-ignore-db options to control binary logging.....You should not use these options to control which databases and tables are replicated. Instead, use filtering on the slave to control the events that are executed on the slave.


and


  To make it easier to determine what effect an option set will have, it is recommended that you avoid mixing “do” and “ignore” options, or wildcard and nonwildcard options.


With these two pieces of information, I'd recommend setting up server B with --replicate-wild-do-table=db_name.%. This is because:


  If you need cross-database updates to work, use --replicate-wild-do-table=db_name.% instead.
  [src]