I'd be dubious about the battery too, but here is a device that measure battery discharge capacity (mAh). http://www.californiasailplanes.com/supertest.html. It says it works with NiCD and NiMD batteries, but it should work for any battery. I'm not sure of the normal load on the battery (mA), but I'd pick the 125mA setting, as to high of a setting could damage the battery.

Here a pricer option:http://http://www.siriuselectronics....roducts_id=348

The basic test method required integrating the battery's current over time. This is most easily done by connecting the battery in series with a current meter / multi-meter and a resistor (appropriately sized to give a reasonable current draw on the battery). Then measure the current over time (shouldn't change much till the battery nears the end of it's life). Plot the data; calculate the area under the curve and you have the battery's capacity in Amp-hours or mAmp-hours (depending on the units used). Assuming the maximum current draw on the battery is 100ma @ 3.7V, you'd need a 37ohm resistor (or something close). If the battery really is 1950mAh & you only drawings 100ma, this test would take 19.5 hours.