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One method is o apply an Offset to the signal so that it never goes negative.I suggest downloading LT Spice form Linear Tech to simulate the circuit. This will quickly answer your questions as to what happens.http://www.linear.com/designtools/software/
What is the signal referenced to? If the input AC signal isn't referenced to anything, then can you tie the "low" end of the signal to ground in your system, and thus have the signal be all positive?If you need isolation, and programmable gain, and offset, the easiest solution is to use a good instrumentation amp, which is like an opamp that's designed for exactly the use case you describe. TI, Analog Devices, Intersil, and others make fine IAs. Try the AD8130 for example. The datasheet shows an output offset/level translator on page 34: http://www.analog.com/static/imported-files/data_sheets/AD8129_8130.pdfDraw-back: Only available in SMD packages, not DIP.
The AD8130 or AD8129 will work great for you, then, as it supports an output voltage offset together with gain. Program it with 100x gain and set the offset to 2.5V and you'll be in business. +15 mV will give you 2.5V + (15 mV * 100) == 4.0V out; -15 mV will give you 2.5V + (-15mV * 100) == 1.0V out. This range should work fine for the ADC input you have.I think this is illustrated in diagram 140 in the data sheet.Soldering a SOIC package by hand is doable if you have a fine-pitch tip for your soldering iron.
I'm assuming you're sampling this using a microcontroller. If so, you can do the additional signal filtering in software.The problem with "full-wave rectified" signals is that any diode will have at least a 0.2 V drop (and likely much more.) Thus, you have to boost the signal to the point where such drop won't matter. If 1% loss is OK, then boost it so the p-p is at least 40 V (more like 100V unless you use very specialized low-drop diodes.)If you need the analog signal out again, then it may still be cheaper to use the AD8130, then an ADC, then digital filtering, then a DAC to generate the signal out.