Probably will, have you got the printer installed on the Windows 7 machine, is it showing up in the "Devices and Printers" Console? Is it hosted by an XP machine as in the OPs situation? If so right click it and choose "printer properties", select the "advanced tab" at the top, then click the "new driver" button the "NEXT" then in the left column select "HP" then in the right column scroll down near the bottom where the "hp psc" printers start and select the "hp psc 1300 series" driver.
*******Ok but no joy as that driver is not listed
I beleive that Windows 7 has built in native driver support for this printer (at least on my 64bit ultimate install it does, see attachment) so see if that works, if not, you may have to resort to physically connecting the printer to the Windows 7 machine and double check the Windows Update Utility to get it to work properly when physically attached,
********Ok, did that, during immediately after I plugged in the usb cable it looked like a bunch of psc drivers were installed.
********I then looked at the printer and it was in trouble mode, trouble shoot resulted in no driver and then obtained a driver for windows update. The usb connected printer now worked on the laptop.
*********However when I go in and try 'new driver' on the network printer although psc 1300 is listed, it is not available. I'm trying to go through the windows update gain for this but it seems to taking an inordinate length of time to populate the list. Will get back to you later.
then you can hook it back to the XP machine and use the method above and when prompted just choose to "keep existing driver".
Good luck, keep us posted.