The problem of the duplicated Ethernet messages has been solved. It turns out that there was not one, not two, but three Ethernet cards installed in the system--and two of them were enabled. Apparently, each of the two active cards received the transmissions. When one of the cards was disabled, the messages were only received once. One clue was that when the message was sent to a single IP, it was only received once. It was only when broadcasting to 256. 256. 256. 256 that the message was received twice.
Thank you for your help in solving this problem