No it won't. £20 for a 600watt PSU? It won't even produce half of that stably. Look for something from OCZ, Corsair, Coolermaster etc. You really really really don't want to be tight when buying a psu. You want a decent one otherwise a cheap one will explode and take the rest of the computer with it.
For an HD6870 a 500 Watt or greater power supply with two 75W 6-pin PCI Express® power connectors is recommended.
The total amperage available on the 12v rail is more important than the total power rating (Watts) of the PSU.