Programmatically scroll a JTextArea to the bottom

I have an application that is sort of a networked console. It allows multiple clients to send messages to it, and displays those messages either in a console instance or in a JFrame with a JTextArea.

I ran into the problem of the JTextArea not scrolling down when new text was appended. (FYI, the JTextArea is in a JScrollPane.) Once the JTextArea contained more lines of text than could be displayed, the new text being appended wasn’t visible unless I scrolled. What I wanted was automatic scrolling. That was accomplished with one line of code:

textArea.setCaretPosition(textArea.getDocument().getLength());