Purée the strawberries in a food processor until smooth.
In a saucepan, melt the sugar with the water. Add the resulting syrup to the strawberry purée. Blend well. Pass the mixture through a fine sieve, little by little, unclogging the sieve of the seeds when needed. Repeat this step a few times. For the clearest syrup, pass it through a piece of cheesecloth.
Pour the syrup in attractive bottles. Chill. Serve over ice cream, pancakes, waffles, on a pound cake or in carbonated water.