I have a similar project (build-in battery charger). It will be for an Arduino BT but I would like that PCB can be used for an Arduino USB.
I found the MAX712/MAX713 circuit: Mixed-signal and digital signal processing ICs | Analog Devices
It can be used to supply the circuit while charging batteries (NiCd or NiMh). You can set the number of cells to charge. But I don't know if it is possible to connect a 9V battery.
There's a lot of schematics on the web, and some with the Eagle CAD files.
For Li-ion, thank you kg4wsv! The Max1551/Max1555 seems really good! Mixed-signal and digital signal processing ICs | Analog Devices
The problem, it seems it can only supply 280mA. I'm not sure, I didn't well-understand the datasheet.
Do you think we can use this component and link it directly to the Arduino BT or to the Arduino USB throw the USB connector?
[Sorry for my english, I hope that I didn't write too much mistakes. (I'm french)]