DataTemplate Query

Question

I have a DataGridTextColum defines as:

Code:

 <DataGrid.Columns>
     <DataGridTextColumn Binding="{Binding FileName}" HeaderTemplate="{StaticResource FileNameHeaderTemplate}" />
 </DataGrid.Columns>

The HeaderTemplate is defined as:

Code:

        <DataTemplate x:Key="FileNameHeaderTemplate">
            <Grid>
                <Grid.ColumnDefinitions>
                    <ColumnDefinition Width="Auto" />
                    <ColumnDefinition Width="*" />
                </Grid.ColumnDefinitions>
                <Label Grid.Column="0" Content="Filename" />
                <TextBox Grid.Column="1" HorizontalAlignment="Stretch" />
            </Grid>
       </DataTemplate>

My issue is the TextBox is not taking up the remaining space (See Image), what am I doing wrong?