Do I need a 64 bit processor on my Mac Pro to run CS6 and Lightroom 4?

I would like to install Adobe CS6 and Lightroom 4 on my 2008 Mac Pro but may not have the right processor to run it.  I know I will have to upgrade my OS X 10.5.8 to at least OS X 10.6.8.  The system requirements specify a Multicore Intel processor with 64-bit support.  I have a Quad-Core Intel Xeon processor.  Does this mean my Mac will not run this software at all or slowly at best?  Will I only be able to use this software if I buy a new computer?

Here is what I have now:

Hardware Overview:

  Model Name:          Mac Pro

  Model Identifier:          MacPro3,1

  Processor Name:          Quad-Core Intel Xeon

  Processor Speed:          2.8 GHz

  Number Of Processors:          2

  Total Number Of Cores:          8

  L2 Cache (per processor):          12 MB

  Memory:          6 GB

  Bus Speed:          1.6 GHz

  Boot ROM Version:          MP31.006C.B05

  SMC Version (system):          1.25f4

  Serial Number (system):         

  Hardware UUID:          A120CDBF-0CB3-56FA-9AF2-16F2938C5F0D

System Software Overview:

  System Version:          Mac OS X 10.5.8 (9L30)

  Kernel Version:          Darwin 9.8.0

  Boot Volume:          Macintosh HD 320

  Boot Mode:          Normal