Right.
You're not going to solve this by fiddling with it or sticking coinage on the arm.
As someone's already pointed out, the constantly-running thing is just the internal overflow discharging into the bowl, so that's not a problem, just a symptom.
Either the ballcock has sprung a leak and isn't buoyant enough, though that's rare enough nowadays, or the valve isn't closing. That can happen through a piece of grit making its way into the system, or simply by failure. You said the property's 10 years old, so my bet is that the valve seal's gone.
My experience of fixing these things is that it doesn't always go that straightforwardly, so be prepared for some measure of hassle. You need to switch off the water supply (in a civilised bathroom, there should be a tap to the khazi supply itself), then figure out how to dismantle the valve arrangement to get to the innards. You're probably best off whipping the whole lot out and toddling down to the plumber's merchants, where they'll look at it, do a few sharp intakes of breath and (hopefully) flog you the correct replacement washer. Then it's back home and reinstall it. Get an assistant to turn on the water while you check the cistern, just in case you haven't done something up properly and several gallons per minute of water starts to flood your bathroom...
