I am trying to amplify the voltage measured across a shunt resistor (Rs). I am using a non-inverting amplifier circuit (as shown in the figure). The expected voltage across the shunt is in the 0-50mV range, and I would like an amplification gain of around 100 (40dB), so I can measure it with an A2D channel with an input range of 0-5V. To achieve this, the resistors I am using are Rf=10,000Ohm and Rg=100Ohm, giving a closed loop gain of 101. The op-amp I am using is a NTE778A. I am powering this circuit using a 12V DC power supply. The problem I am facing is, whatever voltage there is across the shunt, the output voltage always saturated to a value that is (very) slightly lower than Vcc. I am not sure what mistake I am making, either in the construction of the circuit or in my basic understanding of this amplification concept, and any help is appreciated. If this is a common rookie mistake, please do excuse me.
To take Glenn W9IQ's answer and make it more specific, you should check the data sheet for the spec Input Common Mode Range. For this op amp, with +/- 15 volt supplies, the allowed range is a minimum of +/- 12 volts. In other words, it is not spec'ed to operate with the inputs within 3 volts of either supply (at +/- 15 volts). Granted, you are not using +/- 15, but then again the op amp is not characterized for any other power supply, which should be a warning flag. It will probably do OK at 0/15, but it may not. Assuming that it will work, you still should assume that the inputs must be within the range of 3 to 9 volts.
In other words, your op amp is completely unsuited to the use to which you are putting it. You need an op amp with rail-to-rail inputs, or at least with an input range which includes V-. A cheap example of such an op amp is the LM324.
Even if you do this, there is something you need to keep an eye out for. Your diagram shows the low end of the shunt resistor at ground, and the low end of Rg at ground. If your current is high (let's say, an amp or more), and the shunt is not placed directly next to the op amp, the current will flow through other wiring before being referenced to ground, and this will produce an apparent scale factor error. Worse, if there are other high currents running around your system, they may also be detected to some degree. You may see inexplicable output offsets and other weird effects which reflect those unacknowledged currents. Or maybe not. But depending on what else is going on in your system this may be a problem.
Common op amps generally will not perform well near their rails - ground and 12 V in your case. Powering the amp with a 7 volt or higher +/- supply (14 volt total) would likely cure the problem.
Also note that you should set the unused op amp in the package to a gain of one to prevent oscillations that can disrupt your circuit.