Why : In a DC circuit with power supply and resistors, the current is inversely proportional to the resistance of the circuit. If you short circuit the power source then the current is growing to an enormous value. When you switch the multi-meter to ammeter, its internal resistance is 0 Ω , so its like a short circuit. If you then put the probes of the multi-meter directly on to the power source the current grows over 200mA and the internal fuse blows.
Solution : Replace the internal fuse with a resetable polyfuse.
Step 1: Parts
We have chosen a 200mA - 250V polyfuse which is extremely cheap. You can buy 50 pcs for less that 4$
Step 2: Open Up and Replace
2. Bend the polyfuse as in the picture to solder it easier.
3. Put some solder paste on the outside of the socket and solder the polyfuse to the outside of the socket.
I preferred to solder it on the outside because if sometime i wanted to revert my changes (why?) i could easily cut the polyfuse and put the old fuse in its initial place.
The old fuses were 250mA - 250V but i chose to use 200mA - 250V polyfuses for three reasons. First, they are much cheaper than the 250mA, second i couldn't find 250V but only 72V and third my multi-meters never showed anything above 200mA because the display could not handle numbers above 200 so even if i could find polyfuses of 250mA-250V there would be no difference.
4. Close the multi-meter.
Step 3: Final Thoughts
A polyfuse has some resistance which is not 0. You can see how is the resistance in some datasheets, so it is not a perfect solution for everything.
Also to be on the safe side it is better, if you are using your multi-meter with very high voltages, not to retrofit it.
For a school lab though it is better to have meters that working all the time, than meters that are accurate but usually not working. And for statistics, from my last six multi-meters only one fuse was ok, the other five where blown and i have replaced the fuses twice in this semester.
So, Hurray!!! for the polyfuse.