You need to set AutoGenerateColumns to false if you want to define your own columns, otherwise it thinks you want to add extra columns, like this
<DataGrid Name="DG" AutoGenerateColumns="False" >
<DataGrid.Columns>
<DataGridTextColumn Header="Header" Binding="{Binding Path=name}" />
<DataGridTextColumn Header="Header" Binding="{Binding Path=date}" />
</DataGrid.Columns>
</DataGrid>
Hope this helps