Tripod will give you a free account with a mySql database and php support, these tripod accounts are excellent for test beds, though some of the functionality is removed from mysql due to security and the phpbb can seem a little buggy on these accounts it is still handy to have one.

http://www.tripod.co.uk

The host I use is xcalibre, they are a UK company who are cheep and very reliable, their tech support is also excellent, for the purposes of a phpbb forum you will need 1 database this would cost you £10 on top of the hosting with xcalibre.

http://www.xcalibre.co.uk/

I am not sure if xcalibre will host just a database for you, I have never really used any other paid hosts as the service I receive from them is excellent so I'm not sure of or can recommend any companies that provide the service you require.

Goodluck.

RB