Fractional Brownian motion (FBM) provides a useful model for many physical and biological phenomena demonstrating long-term dependencies and 1/f-type spectral behavior. In this model, only one parameter is necessary to describe the complexity of the data, the Hurst exponent (H). The development of accurate estimators for H is a topic of interest in areas such as radiographic image processing and heart rate variability (HRV) analysis. The development of a 1D estimator utilizing the Expectation-Maximization (EM) algorithm is explained; the estimator is designed for a signal model consisting of FBM and additive white noise. The performance of this estimator is tested on simulated noisy FBM data sets, and found to provide more accurate estimates of H than a maximum likelihood estimator for FBM and the detrended fluctuation analysis (DFA) method.