Depends on the ADC. The ADC needs to have a floating or relative mode, so that the ADC ground is biased to be in the middle of the measurement range. Obviously you are going to loose resolution this way.
Commercial dmm use 'multi slope conversion' technique. The measurement is done by using an integrator which generates a linear voltage ramp - this is the 'slope' part. A counter is started at the ramp start and the ramp voltage is compared to the voltage being measured. When the voltages match, the count value is latched (this is the measured voltage value). The voltage ramp is then ramped down and the count is then checked again, this is the 'multi' bit.