When saturated air ascends, water vapor condenses (changes from a gas to a liquid). This releases energy, called the latent heat of condensation. This energy goes into heating the air parcel. Although the ascending air parcel still cools, it doesn't cool nearly as rapidly as it would without the addition of the latent heat from the condensation.
Here's what our parcel at 20 C and 1000 mb would do on ascent if it starts off saturated:
The difference starts immediately as water vapor starts condensing. By the time the air parcel has ascended to 700mb, it has cooled 12.5 C to 7.5 C, which is much warmer than the parcel without the benefit of water vapor. By the time the air parcel gets above the 400 mb level, most of the water vapor has condensed out, so its rate of cooling begins to approach the dry adiabatic lapse rate.
The rate of cooling of an ascending moist air parcel is called, you guessed it, the moist adiabatic lapse rate. But the moist adiabatic lapse rate is not a constant; it's larger (similar to the dry adiabatic lapse rate) at very low pressures and temperatures, and smaller (much less than the dry adiabatic lapse rate) at higher pressures and temperatures.
