Translation:The scale in her store can measure up to one milligram.
Should really be "scales", which means the apparatus used to measure weight. The plural is used in a similar way to scissors and trousers. I guess it comes from the balance type of scales, with two platters pivoting at a central point. Maybe you can also use the singular, but the plural is more common (I'm in NZ).
The singular "scale" means relative size, or the graduations on a ruler.
And... that would be one fancy high-tech set of scales that measured up to one milligram! Probably something illegal you'd be measuring with them. We'd say "down to" one milligram, or "in milligrams" meaning to a sensitivity of one mg. Then maybe you're measuring jewel weights.