I'm afraid the FTDI-option may be difficult.
With bootloader the controller will program the rest of the memory by USB-port (or adapter) since it knows which pins are used for programming. It's more or less a cheap way to use 1 port for both communications and burning the chip without extra equipment. Without... it won't know how to handle the packets sent by your PC.
You'll now have to burn it by its ISP-port.
I did found an article which does use the FTDI in bitbanging-mode to write a new bootloader using the ISP-port, but don't know whether your adapter has the right pins available. This one is also written for the diecimila- arduino. Afaik dweeney is programmed as duemilanove.
I don't know which chip is used in your ardweeney though and getting the right bootloader for the right chip is important. http://www.geocities.jp/arduino_diecimila/bootloader/index_en.html
You could also build an ISP-programmer when your PC has a parallel port. All it takes is some cable and 3 resistors. Since the Ardweeney has no six pin ISP connector, you'll have to connect your programmer to the right pins.
Advantage may be that you could easier choose the right bootloader and burn the bootloader just using the Arduino- software on your PC.
More details can be found here...http://www.arduino.cc/en/Hacking/Bootloader
ISP stands for In System Programming which suggests you could go straight ahead, you probably can, but it may be wise look what's connected to those pins now.