Why can't I get my GUI and KeyListener to work together?

If I do not add any of my GUI components to the JFrame's Container the KeyListener works fine. However, if I add one JButton, for example, the KeyListener no longer works. Why???

import java.awt.*;
import java.awt.event.*;
import javax.swing.*;

public class KeyDemo2 extends JFrame implements KeyListener, ActionListener
{
   private String line1 = "", line2 = "";
   private String line3 = "";
   private JTextArea textArea;
   private JButton button;

   public KeyDemo2()
   {
      super( "Demonstrating Keystroke Events" );

      textArea = new JTextArea( 10, 25 );
      textArea.setText( "Press any key on the keyboard..." );
      textArea.setEnabled( false );
      Container container = getContentPane();
      container.setLayout( new FlowLayout( FlowLayout.CENTER ) );
      button = new JButton("Test");

      addKeyListener( this );

      container.add( textArea );
      container.add( button ); //If I comment out this line the Keylistener works fine.

      button.addActionListener(this);

      setSize( 400, 400 );
      setVisible( true );
   }

   public void keyPressed( KeyEvent event )
   {
      line1 = "Key pressed: " +
         event.getKeyText( event.getKeyCode() );
      setLines2and3( event );
   }

   public void keyReleased( KeyEvent event )
   {
      line1 = "Key released: " +
         event.getKeyText( event.getKeyCode() );
      setLines2and3( event );
   }

   public void keyTyped( KeyEvent event )
   {
       line1 = "Key typed: " + event.getKeyChar();
      setLines2and3( event );
   }

   public void actionPerformed( ActionEvent e )
   {

   }

   private void setLines2and3( KeyEvent event )
   {
      line2 = "This key is " +
         ( event.isActionKey() ? "" : "not " ) +
         "an action key";

      String temp =
         event.getKeyModifiersText( event.getModifiers() );

      line3 = "Modifier keys pressed: " +
         ( temp.equals( "" ) ? "none" : temp );

      textArea.setText(
         line1 + "\n" + line2 + "\n" + line3 + "\n" );
   }

   public static void main( String args[] )
   {
      KeyDemo2 application = new KeyDemo2();

      application.setDefaultCloseOperation(
         JFrame.EXIT_ON_CLOSE );
   }

}