Very overgrown blackberry

We have inherited a very overgrown blackberry at the bottom of the garden. Yesterday I gave it a thorough 'short back and sides' and exposed more healthy unripe fruit but feel it would benefit from a good prune. When does one do this and how much can one cut/hack it back?

I have a similar on my allotment, the more i cut it back the more it grows. I intend to cut it right back when its finished fruiting so in the autumn .I have rampant raspberries all around it that need taking in hand so they will get the chop late autumn.

I'm currently trying to get rid of a load of brambles from my garden (it's TOTALLY overgrown).  You have to be very unlucky (or lucky in my case) to kill them totally, I have been cutting mine back to just above ground level, I went away on holiday and when I got back after a week, those plants I'd cut to ground level had already grown a foot.  Personally, I'd wait for the fruits to ripen, and then cut it back some more.

If/when I get rid of the brambles in my garden, I will NEVER plant them again!

I hope I am not repeating myself as I thought I had already sent a reply - bear with!

Thank you for your replies Maud is in the garden and Mummy Muddy Paws - by the way I love both of your names! I shall take your advice and once the last fruits have ripened I shall be having a good session of dealing with the brute 

You do not say if it is a cultivated blackberry or a wild one.

If you want fruit next year only cut to the ground this years fruiting branches when they have finished fruiting.

If you cut out the non-fruiting shoots you will not get any fruit next year, though you can cut them back a little if they are too long. If you want to propogate new plants pin the end of this year's shoots to the soil and they will root. Then you can sever them from the parent plant.