Amplifying Millivolts with a Single Power Supply Op Amp in a Large Gain

I have a signal around 1mV-10mV to be amplified. This signal should be fed into an ADC (ADS1115 ) which has minimum full scale range of 0.256V. So the gain should be at least around 300. I am thinking of doing this amplification with a single supply (5V) low power Op-amp. I went through some op amps (OPA, LM) and can't find if they are good enough as some don't have libraries to simulate and most are out of stock in our area.
I would like to know if it is possible to achieve such a gain with an acceptable accuracy (error level 5~10%) and some suggestions of particular op amp models. Couldn't find an ADC which has lower than 0.256V full scale range, seeking for suggestions if you have any, so I wouldn't have to amplify the signals in a large factor.

• There are a number of ADCs that might be suitable. A common application is load cells where the values are in the microvolt range. A sigma-delta converter is normally used and these can be around 24 bit resolution. Commented Apr 24, 2021 at 4:09
• A 10mV signal into 256mV range would be a gain of 25.6, not 300. That is very reasonable for an opamp. Commented Apr 24, 2021 at 4:13
• @user1850479 when it is around 1mV it has to be 256 which I roughly said as 300. Commented Apr 24, 2021 at 4:20
• @Kartman I am not familiar with sigma-delta converters and went through some datasheets too. I couldn't find what was their lowest detectable voltage. Could you tell me if there is a particular term for ADCs for the lowest detectable voltage except minimum full scale range. So as your suggestion, would I be able to use a sigma delta converter directly without using an op amp measure my mentioned voltage range (1mV-10mV) Commented Apr 24, 2021 at 4:24
• That is what Kartman is saying. I believe that it is true. There would be no point in making 24 bit ADC's if it was not possible to read them to at least 22 bits or so accurately. But I am sure you will have to pay attention to low-noise design techniques to make it work well. Calibration will probably be needed. This is true for op-amp design also, though. Commented Apr 24, 2021 at 6:26