I wanted an Exim server which could send mail on its own without the use of another SMTP server.

First see this for some general explantion.

I used this config file: 

# /etc/exim4/update-exim4.conf.conf
#
# Edit this file and /etc/mailname by hand and execute update-exim4.conf
# yourself or use 'dpkg-reconfigure exim4-config'
#
# Please note that this is _not_ a dpkg-conffile and that automatic changes
# to this file might happen. The code handling this will honor your local
# changes, so this is usually fine, but will break local schemes that mess
# around with multiple versions of the file.
#
# update-exim4.conf uses this file to determine variable values to generate
# exim configuration macros for the configuration file.
#
# Most settings found in here do have corresponding questions in the
# Debconf configuration, but not all of them.
#
# This is a Debian specific file
 
dc_eximconfig_configtype='internet'
dc_other_hostnames='existingdomain.com'
dc_local_interfaces='127.0.0.1'
dc_readhost=''
dc_relay_domains=''
dc_minimaldns='false'
dc_relay_nets=''
dc_smarthost=''
CFILEMODE='644'
dc_use_split_config='false'
dc_hide_mailname=''
dc_mailname_in_oh='true'
dc_localdelivery='mail_spool'

Also remember to put that existingdomain.com in /etc/mailname.