That's a common offering here in my home city by any company. You can ring them up and agree a price.

On the technology side... Gett.com seem to be offering an identical solution to Uber with the bonus of fixed price service within certain bounds. Looking at trips within a few miles of the city centre it'll offer a fixed price or "by the meter". Anything too far out however and it'll offer only "by the meter". Still, worth a punt?

EDIT: Bit more of a review...

I've contacted their support in the past and it's been pants. A couple of weeks for a human reply.

I also don't like how you have to decide the tip before the trip ends. Once you're dropped off, 'bing' the money is gone. I guess this is a culture difference but I prefer to decide to tip once I've decided I'm happy with the service. it's not a given.