If the difference between the dry bulb temperature and wet bulb temperature is 0°C, what is the relative humidity?

The relative humidity is 100%.  The device you are describing is called a psychrometer, two thermometers, one with a wick on the end of it. 
The two thermometers are attached to a sling, or a stick, devised for the scientist to wet the wick, producing a "wet" thermometer.  The scientist then swings the device for a few minutes, long enough for the air to cause the water in the wick of the wet thermometer to evaporate.  The evaporation produces a cooling effect.  The scientist subtracts the wet bulb reading from the dry bulb reading.  The greater the number of degrees difference, the less water vapor is in the air, which means the relative humidity is lower.  In other words, there is more room for the liquid water molecules to evaporate into gaseous water molecules.  So the relative humidity is lower. 

In this case, however, there was NO difference between the readings of the wet bulb and the dry bulb thermometers.  This means there is absolutely no room at this temperature for water to evaporate, indicating the humidity of the air is saturated, or full.  So the relative humidity is 100%.