Arduino Uno - Analogue Signal Conversion Issues
[For the most part, below is just my conjecture.  Then again, troubleshooting often involves conjectures.]

When you connect the amplifier to the A/D pin, you may be increasing the load capacitance. which the amplifier output is "seeing".  It could be due to the capacitance of the A/D input, or dues to the capacitance of the cable.  This (suspected) capacitive load may be causing your amplifier to oscillate.

We don't know the model of the amplifier (?), so we don't know how much load capacitance it should be able to drive.  We also don't know what you cabling looks like: is it 200mm or 20m?

A quick &amp; dirty fix would be to add as series resistor (say, 1kΩ) between the output of the amplifier and the cable.  I'm calling this a "dirty" fix, because you are increasing the impedance that the A/D converter will see.
A serious fix would be to drive the cable with a buffer.