I have a system that has an old network card, supporting 10 Mbps Half duplex. Is there a way to "wrap" the system with a router or switch that will allow me to connect it to a 1000 Mbps full duplex network?
The router/switch you are plugging into should negotiate with the card. If you are plugging into a managed switch, you will likely be able to hard code the port's speed/duplex. In a Cisco device this would look like so:
Switch# conf t
Switch(config)# int fa0/0
Switch(config)# speed 10
Switch(config)# duplex half
Have you tried simply plugging it in?
Many, even most, (but not all) gigabit ports will negotiate down there.
If the one you have won't, buy a small switch that will (look for specs like 10/100/1000 for the port speeds), and put it between slow old device and the rest of the network.
Alternatively, put a new(er) network card in the device.
I have done this several times (connecting 10M devices to 1G ports) and what I can tell is there is no rule for it, that could tell us in advance whether it will work or not. In theory, it should work, but some devices simply can't negotiate with the switch (or the switch with the device), which means that you will have to force the speed/duplex, just like the answer before.
Just plug in and make a test, just like the in first answer. In case in case it doesn't work, start forcing the speed/duplex and good luck!