We are controlling ants with Diatomaceous Earth. We used this years ago in Texas to control beetles and cockroaches and I had forgotten how well it worked then.
In our current house there are two spots where we have to deal with ants all season. Ant spray gave me a nasty headache so I stopped using that. I also tried to control them with poison, which didn't really work, and ant traps, which also didn't work. Then I just gave up and would sweep them out of the garage every couple of weeks. That also didn't work.
So far this works.
I got Food Grade Diatomaceous Earth on amazon.com, but you can get it anywhere. I got Food Grade so I don't have to worry about any other 'Bonus' ingredients being in it and I can use it by the food garden.
Here is some great background info on using Diatomaceous Earth.
Using White Vinegar to Control Weeds
As far as weed control goes, we have plans to add White Clover, Dandelions and some other productive ground cover and more edible weeds. I'm learning that the best weed control is overgrowing with plants you want to have there. In fact, there are only two areas of the garden where we control the weeds at all; the rest of the weeds we overgrow with (mainly) edible weeds, or just let them do their stuff.
One area of the garden gets no additional water after the spring rain, so these weeds are robust, and living in sun dried clay. A simple mist of vinegar seems to take them down, but I may have to do this on a more regular basis.
It is worth the extra effort to NEVER have to use something like Roundup. But I anticipate long term having to exert less effort.
In the area of the yard where the dogs play we have grass and an unwatered patch that will someday be a deck. In this patch we get weeds that are bad for dogs, like Foxtails, and weeds that produce a lot of seeds that blow into the grass. Since the grass is organic it makes it harder to keep it weed free, so I want to control these weeds and the weeds that popup in the sidewalk around the front grass.
We have started controlling these unruly weeds with a spray bottle of white vinegar. White vinegar is very acidic so don't get any in your eyes and take precautions with gloves if it bothers your skin. I used it to clean a lot of stuff and it doesn't bother me, but you may be different.
I'll let you know how this white vinegar weed thing works throughout the season, but so far so good - as you can see by the pictures from a couple of weeks ago when we started. Next season I will put down seeds for ground cover just before the rain and then I can proactively choose the KINDS of weeds we get. Then they're not really weeds anymore I guess.
The only downside to spraying weeds with white vinegar is that it does start to smell like a tasty wild salad, and then I get hungry...
Other organic weed control methods
Several other areas of organic weed control aside from overgrowing are mowing high, using organic pre-emergents that stop seeds from germinating, and controlling soil pH. An interesting thing I've been learning about soil pH management is that the effect has less to do with the plant itself and more to do with the impact to the bacteria and fungus required to support that particular type of plant in the root system. More on that later.
In the meantime, here is an article by Paul Wheaton with some more great tips, Organic Lawn Care for the Cheap and Lazy.