Hello, I have an Arduino UNO revision 3, I was wondering if I could desolder the USB-B connector on the board and replace it with a Mini USB jack? Thanks.
Not easily.
There is a mini-b connector or 2 with thru-hole 'legs' for mounting the parts.
Here is a comparison of the USB-B vs the USB-mini's.
So you could probably get the mounting holes soldered.
You'll probably have to bend the leads up so you can add wires to run down to the holes.
An order of magnitude easier to buy a cable I think.......
Before the advent of these new fangled Mini and Micro USB sockets, the USB B was standard issue on a lead. 
Pluggy,
You're suggesting remove the USB-B, take a USB Mini cable and cut the end off and wire that to the board?
No, I'm suggesting he buys a standard USB printer lead with the right connectors, plugs one end into the Uno and the other into the PC. Simples.
Blasted comparethemarket.com adverts....
But I think the implied problem is that the USB-B connector is too big and OP wanted more clearance there.
I will bend the leads, I just want to know if you CAN do it and it will work, it´s that here in Mexico USB B cables are expensive, Mini cables are much more affordable.
Yes, electrically it will work.
Personally I think you would be better off cutting the end off of a cable, stripping the leads, and attaching the leads to the boards.
As you can see from the side by side pictures, the mini-b pins will be right over the USB-B pins.
I suppose you could add a piece of heavier wire from the mounting holes and use to secure a non-thru hole mini-b away closer to the side of the board.
For future reference: Replace Mini-B to Micro-B