Success! (At lest in the lab). My soldering job is horrific, but I got it to work. Here's what I did:
Ordered the relay @epfunke linked above. (Thanks for the tip!)
Soldered a ground wire to the PCB ground and a power lead to the #1 pin on the chip in the pic above (which is the power for the #1 LED).
On the relay I connected:
the power lead to the pos. power terminal (DC+, or VCC, depending on your relay)
the ground wire to the neg. terminal, and
a jumper wire from the pos. terminal to the input terminal. This causes the relay to turn on when the speaker's LED pulses on, and off when the LED turns off (which it does once per second for six seconds when the alarm is tripped)
Connected the relay's Normally On and Common terminals to the Ground and Sensor terminals on the I/O Linc. (Doesn't matter which is connected to which.)
Now when the alarm is tripped the I/O Linc sensor pulses on, then off, six times.
To deal with the pulse I programed a State Variable in the ISY to count the number of times the I/O Linc sensor turns on.
I then wrote a test program
If: $Driveway_LED_count >= 4 (I started with "6", but it didn't work right. When I reduced it to "4," it worked perfectly. Not sure why.)
Then: Set 'Chandelier' On; Wait 10 seconds; Set 'Chandelier' Off; $Driveway_LED_count = 0
End result: When I trip the driveway alarm the LED pulses six times, my chandelier turns on, then turns off at appointed time, and the LED pulse count resets to zero.
This is exactly what I wanted. If someone drives or walks down my driveway in the middle of the night, I want my landscape lighting to come on and stay on for about an hour, then turn back off. We live out in the country and deer walk through my front yard all the time. So I don't want an internal alarm going off. This will be enough to make sure people coming up to my house know they are on Candid Camera.
Thanks to all of you for the help and advice. This was fun.