Put Apache, you and that DB file to same group (say group 'www') and make this file 660.
It will be good idea to make apache run as a separate user (say user 'apache') not as nobody. Nobody is used for some other system tasks (NFS,...) and should not be a part of any groups, save for 'nogroup' maybe.

But there is still a slight security risk that some other misfunctioned CGI script can be fooled to read/write this file. This can be solved only by using a setuid/setgid script wrapper.