Custom DataGrid column with a CellTemplate and binding

Question

I need to create a reusable DataGrid column with a custom CellTemplate. This CellTemplate should, among other things, contain a TextBlock control to which I need to bind values to display in the DataGrid. All examples I've seen so far specified the CellTemplate for a column directly when using it in a DataGrid's Columns property and also specified a Binding directly to the TextBlock, e.g.:

<data:DataGrid>
    <data:DataGrid.Columns>
        <data:DataGridTemplateColumn Header="Name">
            <data:DataGridTemplateColumn.CellTemplate>
                <DataTemplate>
                    <TextBlock Text="{Binding Name}"/>
                </DataTemplate>
            </data:DataGridTemplateColumn.CellTemplate>
        </data:DataGridTemplateColumn>
    </data:DataGrid.Columns>
</data:DataGrid>

I need to encapsulate this column into a single, reusable control and I want to specify a Binding for it just like for an ordinary DataGridTextColumn. In other words, when using this custom column, I just want to write something like this:

<data:DataGrid>
    <data:DataGrid.Columns>
        <controls:CustomColumn Header="Name" Binding="{Binding Name}" />
    </data:DataGrid.Columns>
</data:DataGrid>

Problem is that the DataGridTemplateColumn my custom column is inheriting from does not have the Binding property. I thought that I will be able to use DataGridBoundColumn, but it's not possible to specify a CellTemplate for it.

How to achieve a desired behavior? Thank you!