Hello there,
Some op amps will not go all the way to ground when the negative supply voltage is also at ground (0v). There are at least two reasons for this.
One is the input offset voltage, which when multiplied by the circuit gain, may cause the output to be much higher than ground for 0v input.
The second reason (and the one i'll address here) is that the output stage is not designed well enough to reach down to 0v (sometimes the sat voltage causes this).
If the output voltage is caused by the first reason above then you have to supply some input offset adjustment to compensate, which im sure you know.
If the output voltage is caused by the second reason above however then all that has to be done is to drop some voltage so the output goes all the way to 0v.
One simple way to get this to happen is to insert an Si diode in series with the output, with the anode (not the cathode) connected directly to the output of the op amp, and the feedback resistor coming off of the cathode instead of the op amp output, and the load taken from the cathode of the diode. What this fix does is it forces the output of the op amp to be higher than ground (roughly 0.7v) in order to produce a cathode voltage of exactly zero volts. That provides the load circuit with a true zero volts as long as the load does not source too much current.
The other requirement is that another load resistor might have to be added from the cathode to ground in order to properly pull the load down to 0v. This resistor value will have to be chosen properly in order to both get the load down to 0v and also to allow the op amp to still be able to source the required max output voltage, keeping in mind that there is now 0.7v less than before. If there is no resistor that can satisfy both these conditions, then it might be necessary to add an active output stage rather than a simple diode, or another possible solution.
The other possibility is to provide a small negative bias to the negative supply rail of the op amp. This can be done sometimes if the grounds of input and output can be taken from a different ground other than true ground. the other ground is produced by using a diode in series with the negative lead of the power supply where the "true" ground becomes the anode of the diode while the cathode becomes the op amp minus supply and power supply ground. This puts the op amp minus supply rail at roughly -0.7v compared to the analog grounds. This is often just enough to get the output of the op amp to go to a true zero volts. Of course a cap across the diode is a good idea too.
It would help if we knew what kind of load you were trying to drive with the op amp.