A freezer should never be completely disconnected from a mini fridge; that is, it should never be cut off or out. Most mini fridges are cooled by a pipe that draws freon through the freezer. If you cut the pipe from the freezer, you could have a freon leak. If you take the pipe out, the fridge will simply not refrigerate, as there will be no cool air pumping through. Many people, home brewers, for example, want to move the freezer in order to make more room for a larger object. Here is the cheapest, easiest and safest way you can move a freezer without completely disconnecting it.
Find out how the freezer connects to the fridge. Every fridge is different, but the freezers in many are trays that are connected loosely to the top of the inside of the fridge with screws.
Look for the pipe that brings freon into the freezer. Take stock of how long it is, what material it's made of and how easy it looks to bend; don't move it yet.
Make a plan. It's best to know exactly how you want to move your freezer before you move it because you will want to twist the freon pipes as little as possible; they can easily crack.
Detach the freezer.
Duct tape it to the back of the fridge. Since the freon pipe is made of some kind of metal, it will likely help hold the freezer vertical. The duct tape helps to keep it from falling into the middle of the fridge. Secure as much tape as possible to the sides of the inside of the fridge, as this will help give it leverage to keep the freezer back.
Do not disconnect the freezer entirely. The pipe that connects the freon to the freezer is also delicate, so be careful when moving it; moving it too much may cause the pipe to crack, and you will get a leak.