LarsJensen: I take it your Apple mini has a DVI output? You will get the best result if you get a DVI -> HDMI cable and connect the mini to any of the HDMI sockets on the back.
/Lars
And to get digital audio have an optical cable from the mini (headphones socket) to an optical/coaxial converter and then coaxial cable to the TV to one of the available connections.In the Connections Menu You may assign the mini with any available device type (ie DVD, V.MEM, PC etc).
Add also an IR cable to the respective input in the TV and define that input as Apple Remote from PUC (if available in your configuration, might need an update/dealer support). Personally I am using the mini as my primary DVD unit. That is why it is assigned as DVD in the Connections Menu of the BS3. This way you can use Apple remote functions and Frontrow.
This is the way I have connected a mini to my BS3.