However, you can elect to add a vertical scroll bar to the control using the VerticalScrollBarVisibility property. By default, a vertical scroll bar is not present and the TextBox only scrolls if the cursor is moved into a hidden area.
NOTE this answer applies to a TextBlock (a read-only text element) as asked for in the original question.
If you want to programmatically scroll the text within a TextBox control, you can use any of the scrolling methods available to a ScrollViewer. When you provide a multiline TextBox, it is likely that vertical scrolling will be required. Answers: Wrap it in a scroll viewer: