The modern imaging techniques of positron emission tomography and of single photon emission computed tomography are not only two of the most important tools for studying the functional characteristics of the brain, but they now also play a vital role in several areas of clinical medicine, including neurology, oncology and cardiology. The basic mathematical problems associated with these techniques are the construction of the inverse of the Radon transform and of the inverse of the so-called attenuated Radon transform, respectively. An exact formula for the inverse Radon transform is well known, whereas that for the inverse attenuated Radon transform was obtained only recently by R. Novikov. The latter formula was constructed by using a method introduced earlier by R. Novikov and the first author in connection with a novel derivation of the inverse Radon transform. Here, we first show that the appropriate use of that earlier result yields immediately an analytic formula for the inverse attenuated Radon transform. We then present an algorithm for the numerical implementation of this analytic formula, based on approximating the given data in terms of cubic splines. Several numerical tests are presented which suggest that our algorithm is capable of producing accurate reconstruction for realistic phantoms such as the well-known Shepp-Logan phantom.