For your timed minute, you can occupy your brain with a habitual practice like counting. Counting is a habitual activity that takes little thought, and yet, with slight effort, it is enough to keep the brain from wandering.
Practice: Count the seconds of your minute as they pass, focusing on the numbers and no other thoughts. Repeat the count from 1-60 for each minute of your practice.
Practice Variation: Count each inhalation and exhalation of a single breath as a unit. Like this: in one, out one. In two, out two. Continue until you get to fifty. Each time you follow a thought outside the focus of counting, start the count over. If in the process you lose count, start the count over. Surprisingly, this exercise can take you months to master. See also Primary Monkey Mind Meditation.
Purpose: The goal of this method is to break the conscious control slowly through the discipline of using less creative thought. Once the habit of following a thought holds less control, the brain can be educated to step aside when you choose to allow for other mental processes that need less conscious control to perform.