Ultrasound imaging is widely used for diagnostic purposes among the clinicians. A major problem concerning the ultrasound images is their inherent corruption by the multiplicative speckle noise that hampers the quality of the diagnosis, and reduces the efficiency of the algorithms for automatic image processing. In this paper, we propose a new spatially adaptive wavelet-based method in order to reduce the speckle noise from ultrasound images. A spatially adaptive threshold is introduced for denoising the coefficients of log-transformed ultrasound images. The threshold is obtained from a Bayesian maximum a posteriori estimator that is developed using a symmetric normal inverse Gaussian probability density function (PDF) as a prior for modelling the coefficients of the log-transformed reflectivity. A simple and fast method is provided to estimate the parameters of the prior PDF from the neighbouring coefficients. Extensive simulations are carried out using synthetically speckled and ultrasound images. It is shown that the proposed method outperforms several existing techniques in terms of the signal-to-noise ratio, edge preservation index and structural similarity index and visual quality, and in addition, is able to maintain the diagnostically significant details of ultrasound images.