Well, you need to fill the Listbox inside DataGridTemplateColumn. In the itemsource of your Listbox you can add something like this:
Itemsource={Binding {StaticSource domainservice}, Path=Data}
Now, in your DataGridTemplateColumn you can find a DataBinding property (I just cannot remember the real property):
DataBinding={Binding SUBJECT, Mode=TwoWay}
Hope it helps.