First, we simply scraped out as many of the ants and eggs as we could.
We also pulled away all the weeds around the hive so that any new ants are forced to enter the beehive by crawling up one of the table legs. We also sprinkled cinnamon on the ground around the legs of the hives and smeared petroleum jelly around each of the four legs of the beehives to trap the ants. Our hope is that we gave the honeybees enough of a head start to allow them to remove the rest (or at least most) of the ants on their own.
UPDATE: I did not notice any ant activity during a cursory check of the outside of the hives a few days later. (We don't like to bother the bees by opening the hives any more than necessary.)
During the same hive inspection when we were removing ants, we confirmed that we did lose a swarm of bees from the east beehive. A week or so earlier we had seen swarm cells in that hive (capped queen cells that look like peanuts hanging off the edge of the comb). Even though we are still very new to this process, we knew that queen cells in the hive meant that the bees were preparing to swarm. The bees will start to raise a new queen, and before she hatches many bees will take off with the old queen in a swarm until they can find a new home.
In order to allow the bees to follow their swarming instinct but still capture the bees, we needed to identify the existing queen bee and take her and several other bees to a new hive. Unfortunately, after several hours of searching, we were unable to locate the queen.
Fast forward to the hive inspection where we removed ants, and it was very clear that the bees had swarmed. There were significantly fewer bees in the beehive and the queen cells were gone. We will give the new queen some time to mate and start laying new eggs before we bother that hive again to check on her progress.
In other news, we did happen to notice the queen bee while removing ants in the West hive, so it was very encouraging to know that we did have the ability to find a queen after all. Below is a picture of us pointing out the queen to our boys.