Custom DataGrid column with a CellTemplate and binding

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!