No, you can't duplicate a style in the CSS Styles panel in Current mode. The option is also greyed out in Dreamweaver CS6.
The simple workaround is to select Go to Code from the context menu. That opens the style sheet in Split view and positions your cursor inside the selected rule. You can then easily copy and paste the rule directly in the style sheet.
Thanks David. That's a very good tip and one which I had not spotted. Still seems strange that the duplicate option is greyed out but this is a great workaround.
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I think the reason Duplicate is greyed out is because Current mode shows only styles that affect the current selection. When you choose Duplicate, you're prompted to create a new selector, so it would no longer apply to the current selection.
However, if you think it would be useful to enable Duplicate in Current mode, submit a feature request through the form at http://adobe.ly/DWBug.
There could be some logic to it as you say. But I can't see that it would cause any problems to be able to duplicate a current style and it is still possible to create a new style while in current mode and that uses virtually the same dialogue. I think I will follow your suggestion and put in a request. Many thanks for the advice.