The movement of a particle described by Brownian motion is quantified by a single parameter, D, the diffusion constant. The estimation of D from a discrete sequence of noisy observations is a fundamental problem in biological single-particle tracking experiments since it can provide information on the environment and/or the state of the particle itself via the hydrodynamic radius. Here, we present a method to estimate D that takes into account several effects that occur in practice, important for the correct estimation of D, and that have hitherto not been combined together for an estimation of D. These effects are motion blur from the finite integration time of the camera, intermittent trajectories, and time-dependent localization uncertainty. Our estimation procedure, a maximum-likelihood estimation with an information-based confidence interval, follows directly from the likelihood expression for a discretely observed Brownian trajectory that explicitly includes these effects. We begin with the formulation of the likelihood expression and then present three methods to find the exact solution. Each method has its own advantages in either computational robustness, theoretical insight, or the estimation of hidden variables. The Fisher information for this likelihood distribution is calculated and analyzed to show that localization uncertainties impose a lower bound on the estimation of D. Confidence intervals are established and then used to evaluate our estimator on simulated data with experimentally relevant camera effects to demonstrate the benefit of incorporating variable localization errors.