Arduino with atmel tools
I'm pretty sure that you can.  All the schematics for the AVR910 programmer that I have found show it with a 6 pin interface.  

10 Pin to 6 Pin Converter

A converter ought to do it.  Some programmers come with both interfaces, for instance the Adafruit Attiny USB programmer.  Loooking at its schematic, I can see that the 10 pin programmer has the same connections on it as the 6 pin, just a few not connected pins.  I assume that the 10 pin connectors pinout is standardised.  You of course could convert it yourself or build a similar converter as above without too much difficulty.