Jibberish on hardware serial port

I am having a problem trying to interface to the hardware serial port on my duemillanova. I thought at first that it was some problem with my program sketch so I went back to a very basic program and used the defaul "ASCII Table" program in the sketchbook>examples>communication folder.
This program seemingly works correctly when viewed on the Serial Monitor. The program defines the port to be 9600 and if I set the monitor to 9600 everything shows up fine there.

However, if I connect a cable from digital pin1 (TX) and digital pin 14 (GND) to another computer and open hyperterminal there, setting 9600, N,8,1,off I get a lot of characters come through but they are all garbage. The interesting thing is that they are exactly repeatable. garbage. Every time I press the reset button on the board to have it run through again, I get the exact same string of jibberish on the hyperterminal, even though the right stuff shows on the Serial Monitor.

Obviously, I am doing something wrong, but I have tried playing with baud rate, parity, etc. on the hyperterminal to no effect. I even wonered if the ground was wrong, and tried using the Gnd connection on the Power header, to no avail. Does anyone have any idea of what I am doing wrong in simply trying to do simple serial communications to another computer?

if I connect a cable from digital pin1 (TX) and digital pin 14 (GND) to another computer and open hyperterminal there, setting 9600, N,8,1,off

What are you using for RS232 level conversion/inversion?

OK, there you may have hit the nail on the head, I have no idea what you are talking about!
I was under the impression that if you used the hardware serial port on the arduino you didn't need to do anything more than what is included in the ASCIITable sketch. Am I wrong? What else do I need to do?

From some things I read after your post it looks like the problem may be TTL vs RS-232, no? I need to convert the 0v & 5v nominal TTL signal from the Adruino to –12v & 12v RS232 serial signal. Is that it?

Briefly:
The pins at 0 and 1 use ordinary logic levels where 0V is a low and 5V is a high.
RS232 (nominally) uses +12V to represent a low, and -12V to represent a high.
So, if you want to use pins 0 and 1 to talk to an RS232 device, you need to be introduced to MAX232.
A simple Google search should give you lots of tips.
Apart from the soldering iron one you'll probably need.