Check the data sheet section 6.2 Absolute Maximum Ratings. Either the Vdd-Vss rating of 4.0V or the "Input voltage on any other pin" rating has been violated, causing permanent damage to the device.
If you sent this board to a Failure Analysis lab, they could etch a hole into the epoxy that encapsulates the IC, and probably show you a photograph of where the overvoltage condition caused metal migration or other permanent damage to the IC. This isn't fixable. What the FA lab would be able to tell you basically is, it looks like someone applied too much voltage -- and you already know that.
As Leon Heller commented, the board should be scrapped and replaced. Not only the ARM itself, but very likely most of the other ICs will have been damaged. Even if you tried to work around the overheating problem, there's no longer any guarantee that IC still behaves like an ARM chip. What if its ALU sometimes makes incorrect results or some of its registers sometimes fail to store data. This board is no longer reliable and should be scrapped.
For what it's worth, mistakes like this do happen -- one of the hazards of working on bare prototype PCB instead of nicely packaged finished consumer goods. In this case it should be easy to order a replacement.
First the programming voltage, it is given in the data sheet under the operating conditions of the memory (page 63):
Vprog Programming voltage 2 - 3.6 V
So if I read your suggestion correct it might have been powered only by that coin cell during programming? This might be very very bad indeed. The controller draws a maximum current during write operation (just the flash) of 7 mA. A small coin cell really doesn't cope well with higher currents and it's voltage as well as the capacity dropped probably really fast. Given the datasheet of a CR1220, the voltage drop on a full battery would be at least 0.5 V if you draw 7 mA from it, so it starts out at about 2.35 V, not much headroom.
But I also wonder if your circuit is fine now - have you measured the current it draws and checked all the voltages and their stability if you draw current (connect a resistor decade and vary it a bit)?
Now a possibility on how you might be able to salvage the part. I guess it won't work as you have no trouble connecting to the device, but anyways:
The F103 has a built in bootloader which is accessible if you pull the BOOT pin high during power up. You will also need access to UART1 to be able to communicate with the bootloader. Have a read of section 3.4 of the reference manual. Another good resource is this document.
If you can connect to it, there is a tool (STSW-MCU005STM32 and STM8 Flash loader demonstrator (UM0462)) available from ST which allows to interface with the bootloader and erase or program the chip.
Best Answer
Simple answer, use free CoFlash, it will do the trick. STM Discovery F4 has everything you need on board, all you have to add is USB cable.