How do I prevent the datagrid from displaying its append row (the row at the end with an asterisk)
The DataGrid class does not have a property that controls whether a new row can be added. But the DataView class does have such a property (along with some others such as AllowEdit and AllowDelete). Here is code that will turn off the append row by getting at the dataview associated with the datagrid.
If your datagrid contains links, then Matthew Miller suggest adding Navigate handler such as the one below to disallow the AddNew.