Thanks Daniel,
Your advice helped! Here's the code from my Perl program, and it seems to be working fine.
print MAIL<<EndMAIL;
MIME-Version: 1.0
Content-Type: multipart/alternative; boundary="06986E0E1E196312E032AFBC"
This is a multi-part message in MIME format.
--06986E0E1E196312E032AFBC
Content-Type: text/plain; charset=us-ascii
Content-Transfer-Encoding: 7 bit
$plain_text
--06986E0E1E196312E032AFBC
Content-Type: text/html; charset=us-ascii
Content-Transfer-Encoding: 7bit
$html_text
--06986E0E1E196312E032AFBC--
EndMAIL
;