Hi there,
Li-ion cells are a little more problematic than NiMH if they are not handled properly. They must be charged correctly and can not be discharged too much or they will get ruined. NiMH cells are a little problematic too in that they really need the correct charging method, but they are safer to use overall.
An Li-ion cell requires a charge algorithm similar to a lead acid battery. Luckily, there is an easy way to get this with fairly common parts. For example you can build a Li-ion charger from a common LM317 voltage regulator with a few parts and charge the Li-ion cell fairly accurately. You do need to get the max voltage setting correct however, or at least set it close to the max.
The circuit looks like a standard LM317 voltage regulator circuit, with one exception: there is a small power resistor in series with the input. The power resistor is there to limit current while the cell voltage is under about 4.200 volts.
The drawback to this or any other linear circuit is that it isnt that efficient for voltages far above the output voltage, but then again many other Li-ion charging chips work like this too (although their max input voltage is not very high to begin with).
It's also a good idea to set the max output to about 4.15v instead of 4.20v just in case your meter doesnt read correctly.
If you'd like to see a circuit drawing i can post one with all the details.