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We present an improved method for making pump-and-probe optical measurements in the picosecond to nanosecond time range. In this type of measurement, a pump light pulse is used to excite the sample and the resulting changes in the optical properties are investigated by means of a probe pulse that is time-delayed relative to the pump pulse. In most measurements of this type, a mechanical stage is used to introduce the variable time delay of the probe pulse. As a result of imperfections in the stage motion, alignment problems, and divergence of the probe beam, it has been difficult to make accurate measurements when the time delay of the probe relative to the pump is in the range above a few hundred picoseconds. To overcome these difficulties, we have developed an apparatus that utilizes a single-mode optical fiber. In order to demonstrate the performance of this system, we present results of experiments in which the flow of heat from a thin film into a substrate has been measured.