[
I do not mean an additional A/D converter to read
the value back (althought this would be a nice option, too) but to read
the current digital value that is set to the output.]
You already have the ability to read back the digital value that's feeding the D/A converter. Simply wire the value to an indicator and read it back with your host VI.
The 7831R already has analog in and out, so, technically, you have the ability to loop your analog output back to an input. But there's no point in that since you'll just end up with the same value that's already accessable to you through an indicator.