This module uses the PT2399 digital echo chip from Princeton Electronics. Small Bear Electronics
has them for $3.95. The circuit is based on the Surround Sound/Delay example in the datasheet.
The delay time is variable from about 33mSec to 300mSec by varying a resistance. The datasheet specs a maximum 300mSec delay at 1% distortion, but I found I could go to about 2 seconds with acceptable results.
I implemented voltage control by using an LDR (yellow LED and CDS photoresistor) for the resistive element. A true vactrol could also be used. You want the resitance to vary from as low as possible (for short delays) up to about 100K. Some experimentation will have to be done with the driving current levels for the LED. Make sure the the LED - photoresistor assembly is in a light tight container, since ambient light will have an affect at the low light / high resitance / long delay end.