When a liquid approaches its glass-transition temperatures T(g), the structural relaxation time tau dramatically increases. This basic feature is ubiquitous, but this increase of tau can be classified between strong and fragile extremes using T(g) as a scaling parameter. Liquids, whose tau obeys the Arrhenius law, are called "strong," while "fragile" liquids have the super-Arrhenius behavior. Here we report the first continuous control of the fragility of liquid of the same material over a wide range of fragility, using a continuous liquid-liquid transition. Our study clearly demonstrates that the fragility is not a material-specific quantity, but is controlled by the order parameter governing the liquid-liquid transition, which may be the fraction of locally favored structures in the liquid.