No, your PSU will die a fiery death.
You need a QUALITY 450w+ PSU from Corsair, PCP&C, Enermax, Seasonic or the Antec Earthwatts line.
ATI Radeon™ HD 4850 System Requirements
450 Watt or greater power supply with 75 Watt 6-pin PCI Express® power connector recommended (550 Watt and two 6-pin connectors for ATI CrossFireX™ technology in dual mode)
A 4850 will be much, much better than your current 8600GT.
As cheap as they are now, why put off the upgrade.