Yes, I know Swing & Processing don't play well together, but I had a need for hyperlinks in a text box, so here is some code I've put together that tries to address the HeavyWeight / LightWeight problem.  It also doesn't use a separate frame, so you don't have to worry about trying to capture keystrokes and forcing a window to be "AlwaysOnTop" every few frames.

Love to see improvements if someone wants to hack at it.  In this instance, I used a JEditorPane on a JScrollPane and added HyperLink support.

UPDATE: Improved version in my next post below

1. import javax.swing.*;
2. import processing.opengl.*; 
3. import java.awt.*;
4. import javax.swing.event.HyperlinkListener;
5. import javax.swing.event.HyperlinkEvent;
6. import javax.swing.text.html.HTMLEditorKit;
7. import javax.swing.text.html.HTMLDocument;
8. import javax.swing.text.Document;
9. import javax.swing.text.DefaultStyledDocument;
10. import java.awt.Insets;
11. import java.awt.event.WindowAdapter;
12. import java.awt.event.WindowEvent;
13. import java.awt.event.WindowListener;
14. int position = 0;
15. SwingGUI gui;
16. void setup() {
17.   size(550, 400, OPENGL);
18.   gui = new SwingGUI(this, 100, 100, 300, 200);
19.   gui.setVisible(true);
20.   String myHTMLString = "";
21.   for (int i=0;i<50;i++) {
22.     myHTMLString += "<font color=white>This is a test of a JEditPane inside a JScrollPane within Processing using OpenGL.  Click my Hyperlink for </font><font color=#FF8040><a href=\"http://www.google.com\">Google</a></font><br/><br/></font>";
23.   }
24.   gui.setText(myHTMLString);
25. }
26. void draw() {
27.   background(100);
28. }
29. // An inner class for all GUI elements.
30. class SwingGUI {
31.   int x, y, w, h;
32.   JScrollPane scroll;
33.   JEditorPane panel;
34.   PApplet parent;
35.   boolean visible;
36.   JPopupMenu jm = new JPopupMenu();
37.   /** Creates new form GUI */
38.   public SwingGUI(PApplet parent, int x, int y, int w, int h) {
39.     this.parent = parent;
40.     scroll = new JScrollPane();
41.     panel = new JEditorPane();
42.     try {
43.       // panel.setPage(url);
44.       Insets iSet = new Insets(5, 2, 5, 5);
45.       scroll.setBorder(null);
46.       scroll.setViewportView(panel);
47.       panel.setEditorKit(new HTMLEditorKit());
48.       panel.setContentType("text/html");
49.       panel.setMargin(iSet);
50.       panel.addHyperlinkListener(new MyHyperlinkListener());
51.       panel.setEditable(false);
52.       panel.setBackground(Color.black);
53.       panel.setForeground(Color.white);
54.       //
55.     }
56.     catch (Exception e) {
57.     }
58.     jm.setDefaultLightWeightPopupEnabled(false);
59.     jm.add(scroll);
60.     setLocation(x, y);
61.     setSize(w, h);
62.   }
63.   void setVisible(boolean visible) {
64.     this.visible = visible;
65.     jm.setVisible(visible);
66.   }
67.   boolean isVisible() {
68.     return visible;
69.   }
70.   void setSize(int w, int h) {
71.     this.w = w;
72.     this.h = h;
73.     jm.setPopupSize(this.w, this.h);
74.   }
75.   void setLocation(int x, int y) {
76.     Point p = parent.frame.getLocation();
77.     this.x = p.x + x;
78.     this.y = p.y + y;
79.     jm.setLocation(this.x, this.y);
80.   }
81.   void setText(String textstring) {
82.     try {
83.       InputStream is = new ByteArrayInputStream(textstring.getBytes("UTF-8"));
84.       panel.read(is, "html");
85.     }
86.     catch (Exception e) {
87.     }
88.   }
89.   class MyHyperlinkListener implements javax.swing.event.HyperlinkListener {
90.     public void hyperlinkUpdate(HyperlinkEvent evt) {
91.       if (evt.getEventType() == HyperlinkEvent.EventType.ACTIVATED) {
92.         getlink(evt.getURL().toString());
93.       }
94.     }
95.   }
96.   void getlink(String url) {
97.     link(url);
98.   }
99. }
100. void keyPressed() {
101.   println(key);
102. }