Disable DataGrid current cell border in FullRow selection mode

I am using a DataGrid in row selection mode (i.e., SelectionUnit="FullRow"). I simply want to remove the border that is being placed around the current cell when the user highlights a row in order to have true full row selection (and no cell level selection). I don't mind the notion of the grid maintaining the current cell, I just want to remove that pesky current cell border, perhaps by changing the style of the current cell. What is the easiest way to do this?

You could set the BorderThickness for DataGridCell to 0

<DataGrid ...
          SelectionUnit="FullRow">
    <DataGrid.CellStyle>
        <Style TargetType="DataGridCell">
            <Setter Property="BorderThickness" Value="0"/>
            <!-- Update from comments.
                 Remove the focus indication for the selected cell -->
            <Setter Property="FocusVisualStyle" Value="{x:Null}"/>
        </Style>
    </DataGrid.CellStyle>
    <!-- ... -->
</DataGrid>