In addition to the ants and aphids, we noticed a whole bunch of other strange bugs crawling on the tree (as shown in the photo). After some research, we found that these were lady beetles (also known as ladybugs) in a larva stage. Wikipedia has some good pictures of the various stages (early larva, mid-larva, pupal, and adult), all of which we have seen on our tree. They showed up because they are predators of aphids.
In addition to letting the lady beetles do their thing, we are doing our part to combat the aphids. We borrowed a power washer and have been spraying the leaves with water from our rooftop and from the top of a ladder. This knocks many of the aphids off the leaves, and seems to be washing some of the honeydew and mold off. We tried adding a castille soap additive (which will kill aphids on contact), but we couldn't get the power washer to spray that with enough velocity for some reason. While spraying a big plume of water into the tree from my rooftop with a pistol wand feels almost like I'm acting out a scene fromGhostbusters, it gets old pretty quickly (particularly when the wind starts blowing the water right back at me). We've done it once a weekend for the last three weeks, and hear that six weeks may do the trick.
Now that we have a plan, we hope to jump on the aphids early next year and head off the problem before it gets bad.