I take it the displays you intend to use are both DVI only?
You are right that digital display connections don't usually like to be passively 'split'. Since computers usually use an active two way connection to their displays, this can confuse things. Are the two displays identical?
DVI as a standard can deliver digital or analog signals, but you can't 'split' one signal that's being sent into one analog, one digital. If the displays were identical, you might be able to get away with using a passive splitter to mirror the same desktop to both displays. The only issue then is HDCP authentication. That might not be necessary and might not come into play though. If your mini really is a 1.83 ghz Core 2 Duo, you should have a full sized DVI port. Those minis came with a DVI to VGA adapter. VGA is analog only. If you did use VGA, you could much more easily just split the output to mirror to two displays, as an analog signal doesn't have the same authentication issues. You would need both displays to support the same resolution and timing frequency. If both of your displays support VGA, this is probably your best way to go.