Wash and dry the cucumbers. Using a sharp knife or a mandolin, slice the cucumbers thinly and place in a colander. Sprinkle with salt and toss to coat. Place the colander over a bowl and allow it to sit, covered, for about 1 hour. Rinse off the salt and dry the cucumber slices well. Place them into a sterilized quart jar.
In a small saucepan add the remaining ingredients. Stir to dissolve sugar and bring to a boil. Remove from heat and allow it to cool. Pour the brine over the cucumbers in jar. Cover and refrigerate at least 4 hours, preferably overnight. Will keep, refrigerated, for about 2 weeks.