You can put down cardboard between the rows with a mulch on top of the cardboard. Please don't spray the garden with a weed killer. We use old grass clippings as mulch. What few weeds we get pull very easily.

Most tarps let through way too much light Cardboard seems to be the unanimous choice. You might also try black plastic or even clear. With clear you want to make sure the area is thoroughly watered, tuck it in real well, and do it in the middle of summer --- you lose a season, but it solarizes the soil and kills anything a foot down. Don't worry about the worms and microorganisms.. they'll come back.

Also, to keep weeds down to a minimum, after planting your garden, mulch heavily. The few weeds that may come will be poorly anchored and can be pulled with a slight tug.

kevin