Exim Authenticate With Pam Library On FreeBSD (March 28, 2013)

You need to run exim as root. So edit Local/Makefile thus:

# FIXED_NEVER_USERS=root (comment out)

In your configuration file set:

exim_user = root
exim_group = mail

And, hey presto, it works! I hope it will save you the many hours it took me to figure it out.