Hooray!  I now have both of my LCD's working in Ubuntu, ala "TwinView."  I attemped to use the nVidia confiuration utility, but ended up needing to manually edit my xorg.conf file to get the right settings.  I think the utility would have worked if my secondary LCD was connected using a DVI cable, but unfortunately it only supports VGA so I think the resolutions were not getting auto-detected properly.

For future reference, here's the relevant portions of my xorg.conf file with the dual monitor stuff in bold:

Section "Monitor"
    Identifier     "Acer AL2216W"
    HorizSync       30.0 - 82.0
    VertRefresh     56.0 - 76.0
    ModeLine       "1680x1050" 146.2 1680 1784 1960 2240 1050 1053 1059 1089 +hsync -vsync
    Option         "DPMS"
EndSection

Section "Device"
    Identifier     "nVidia GeForce 8600 GT"
    Driver         "nvidia"
EndSection

Section "Screen"
    Identifier     "Default Screen"
    Device         "nVidia GeForce 8600 GT"
    Monitor        "Acer AL2216W"
    DefaultDepth    16
    Option         "UseFBDev" "true"
    Option         "RenderAccel" "true"
    Option         "TwinView"
    Option         "MetaModes" "1280x1024, 1680x1050; 1024x768, 1280x1024"
    Option         "TwinViewOrientation" "LeftOf"
    SubSection     "Display"
        Depth       16
        Modes      "1680x1050" "1280x1024" "1280x960" "1280x800" "1152x864" "1024x768" "800x600" "640x480"
    EndSubSection
    SubSection     "Display"
        Depth       24
        Modes      "1680x1050" "1280x1024" "1280x960" "1280x800" "1152x864" "1024x768" "800x600" "640x480"
    EndSubSection
EndSection