Finish coat popping off

I am a General Contractor in Missouri and hired a sub-contractor who specializes in stucco work. The house we are working at is approximately 80 years old.

The thickness of the existing concrete stucco is 1" and the texture is the old style that looks kind of like a half moon shape that concaves inward somewhat. The sub-contractor was hired to install new stucco on part of the garage and was to match the existing stucco. They hurried to complete the job in 1 day with the temps around 40 degrees. I know that normally it should take more than 1 coat. The thickness they installed on the new sections is ¾" thick and looked like snowballs instead of the old style of half moons. They also told the homeowners and myself that traditional stucco is all done at one time.

After 3 months and many arguments later, they came back to fix it so the stucco would match the existing texture and at this time they applied an additional ½" on top of the ¾" that was already there. It has been 6 months and the second layer installed is now pulling apart, cracking and falling off in the one section of the garage. I told them they needed to tear out this section and start over. They say they only need to drill holes and patch the areas and will only guarantee it for 1 year. My concern is if this one area is already having problems after 6 months what will happen to the rest of the garage.

What do you suggest I have them do to fix his problem correctly?

I have anwered this question before, so I feel a little redundant. I can tell you how to cure the problem, and possibly how to prevent it in the future.
The new finish will need to come off. White portland won't bond to white portland or a slick basecoat. That is with out a chemical bonder. I have discussed other methods of getting mortar to bond to unpainted stucco, painted stucco and concrete. Without a method like using an acrylic bonding admixture, mortar will always pop off. Sometimes the next day, sometimes after a few weeks, but it will pop off.
I am not dismissing paint on bonders, that is weld-crete. Weld crete works on unpainted stucco, like here, and on concrete. Other bonders, like plaster weld will fail if used outside. Weld crete can fail on painted surfaces, and I have seen weld crete fail when used outside, possible due to using wet paint brushes or rollers, adding water to the weld crete, or just cold temperatures. Bonding admixtures are more reliable, and insurance that the mortar won't fail.
I have discussed this method here.The new finish must be removed. The finish my have a few stubborn areas that don't want to come off, but they can be knocked off.
The next step is to put on a basecoat, at least thick enough to cover the original finish. We use flex-con, an acrylic bonding admixture mixed half and half with water. The formula for the basecoat is just like our usual basecoat, One bag portland type one, one fourth bag of lime, and twenty shovels of sand. The flex con allows mortar to be applied thin, retaining liquid in the mix, preventing dry-outs. Dry outs are where mortar dries before it sets.

Flexcon is mixed half and half with water. Even though it is not necessary, we dry mix the basecoat in a box and then mix the basecoat with the wet mix with a drill, just like our finish coat. We found the mortar spreads easier this way when using acrylic:

Mortar with acrylic mixed with a drill.

If the surface is fairly flat, the mortar can be trowel on and roughed up with a float or a stiff brush. If the surface is real rough, ile here, the mortar can be troweled on thick, and rodded off. (That is, straighted with a straight edge, called a rod) :

Rough areas rodded off.

Finish coat mixed just like in my video link above. Here is your half moon:

Half moon texture.

Prevention is worth a pound of cure, but incompetent people are constantly getting work. Sometimes, they honestly believe they can do stucco. Sometimes, unqualified EIFS workers are called in. I tried contacting a pay per lead service. The only contractor that said he can do stucco would not provide me with and address of a job I could look at. It is a good idea to visit a job the prospective contractor has done. If they don't want to give you and address, get ready to run

Hope this helps. The important thing is to find experienced people to do this for you. It is an age old problem that people say they can do stucco, but can't.