Are you saying that when you set the pot to less than 1.3V, the output voltage remains at 1.3V regardless of how much lower you set the pot? What does the output circuit attached to the OP-AMP look like?
With the given info, I suspect that the load connected to the OP-AMP is sourcing some current that the OP-AMP must shunt to ground. However, the OP-AMP's negative power supply input pin is connected to ground. Therefore, its ability to drive low is limited. I think the output stage of the op-amp cannot pull the pin any lower given the strength of its output driver.
Look at the datasheet of the OP-AMP, it will say how close to the power rails the OP-AMP is able to drive. The solution is to either move the OP-AMP's negative supply terminal to a voltage below ground or change out the OP-AMP for a device that is capable of rail to rail operation.