This how to describes the process for copying a web part.

1. Open a SharePoint site.
2. Make sure you're in Shared Mode.
3. Enter design mode
4. On the web part to copy, click the down arrow on the right and select export.
5. Save the file on your system
6. Go to the site where you want the control added
7. Enter Shared Mode
8. Select Add Web Parts-Import
9. Browse to the file and click import
10. Drop the web part on the page.

You've now copied a web part from one web page to another.  Note that this won't work for ListViewWebParts or ListFormWebParts.