Can I charge à rechargable battery directly from à power source?

Can i charge à rechargable battery directly from à power source without going throughout à charger?

I have bluetooth speaker which is charger using à mini USB port. The USB plug has been destroyed and is no longer usable. Taling the speaker apart I can see where the battery attaches to the PC board. If I 'tap in' to the wires going from the battery to the board and attach a 5V 1A power source, everything seems to work OK. Also, it appears as though the battery charges up.

I could simply cut the battery wires and run the speaker directly from the power block. The question is, if I keep the battery connected will it overcharge and cause problems (such as overheating and catching fire)?

Thanks.