Can I use one battery to power up two circuits?
Electrically, there is no problem running both devices off of the same battery.

However Zinc-carbon 9v batteries only have a capacity of 400 mAh, and Alkalines are not much better at 565 mAh, so neither is going to power your BeagleBoard much more than an hour or so (you can't run the battery all the way down).  And that is not taking into account the original circuit you want to use the battery for, with an unstated current consumption.

Instead you should switch to a higher capacity solution, such as using six high-capacity AA Alkaline batteries with a capacity of up to 2600 mA, over six times the capacity of the individual 9v Zinc-carbon battery.

A better option, if the two devices do not need to be portable, is to use a 9v wall-wart (be sure it's regulated), or use the one designed for the BeagleBoard and run your original device off the 9v battery.