Command+v to copy and then command+f will paste in front (can be found in the edit menu).
If you are copying a layer, select the layer, click and drag it into the new layer icon at the bottom of the layers pallet.
Also if you copy and then paste, as long as your art board is fit to the window it should paste in the center.
If you don't mind grouping, then select and group your layer (click to the right of the layer name in the panel - to the right of the little circle there to select everything on a layer...), then use the align panel with the "align to artboard" button pressed, then ungroup your layer.
[edit - I'm still using CS3, so I don't know how this might get complicated in CS4...]
Hmmm... cut and paste does not seem to center the object on the artboard.
AH! This seem to work. Pressing the center align buttons does align the grouped object to the center of the artboard.