Convert DC 0 to +1V signal to -1 to +1 signal
Use an op-amp with dual-supplies, such as a rail-to-rail input AND output general purpose op-amp, with +-1.8V or +-3.3V supply rails. 

Make a "summing amplifier" configuration, and offset ("add") the input signal with -0.5V, so the op-amp subtracts 0.5V from the input signal, making the output -0.5V -> +0.5V. You may use a resistor divider from the -ve supply rail to generate a -0.5V reference offset.

Next, with the output of the diff-amp, use a voltage follower/buffer op-amp (again, with dual supply rails) with a gain of 2, so the input is multiplied by 2. 

This will result in the +-0.5V range of the difference amplifier becoming +-1V. 

You did not state the bandwidth requirements for this application, so I assume a few 10's of KHz, and my advice here is relevant.