[exim] pipe transport woes

I have a very simple shell script written to handle email messages to
a specific address. But I've noticed some weirdness with using
scripts from Exim.

Let me give a quick example:

foo.sh
------
#!/bin/sh
date >>/tmp/script.log
id >>/tmp/script.log
cat - >>/tmp/script.log
exit 0

When executed from an Exim transport, the script.log file contains
only the results from "id"...
------
uid=1015(dchoma) gid=1015(dchoma)


Where's the date and the message? The script does what it's supposed
to if run from the shell...
------
Sat Dec 16 23:45:23 PST 2006
uid=0(root) gid=0(root) groups=0(root)
this is some sample text


I've read through the documentation several times. I've tried the
"use_shell" option to see if that makes a difference (it doesn't).
What am I missing?