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As the temperature of a cooling object decreases as it relaxes to thermal equilibrium, it is intuitively assumed that a hot object should take longer to cool than a warm one. Yet, some 2,300 years ago, Aristotle observed that "to cool hot water quickly, begin by putting it in the sun". In the 1960s, this counterintuitive phenomenon was rediscovered as the statement that "hot water can freeze faster than cold water" and has become known as the Mpemba effect; it has since been the subject of much experimental investigation and some controversy. Although many specific mechanisms have been proposed, no general consensus exists as to the underlying cause. Here we demonstrate the Mpemba effect in a controlled setting - the thermal quench of a colloidal system immersed in water, which serves as a heat bath. Our results are reproducible and agree quantitatively with calculations based on a recently proposed theoretical framework. By carefully choosing parameters, we observe cooling that is exponentially faster than that observed using typical parameters, in accord with the recently predicted strong Mpemba effect. Our experiments outline the generic conditions needed to accelerate heat removal and relaxation to thermal equilibrium and support the idea that the Mpemba effect is not simply a scientific curiosity concerning how water freezes into ice - one of the many anomalous features of water - but rather the prototype for a wide range of anomalous relaxation phenomena of broad technological importance.