A novel algorithm is presented for the estimation of spacecraft attitude and angular rate from vector observations. Belonging to the class of Monte Carlo sequential methods, the new estimator is a particle filter that uses approximate numerical representation techniques for performing the otherwise exact time propagation and measurement update of potentially non-Gaussian probability density functions in inherently nonlinear systems. The paper develops the filter and its implementation in the case of a low Earth orbit (LEO) spacecraft, acquiring noisy Geomagnetic field measurements via a three-axis magnetometer (TAM). The new estimator copes with the curse of dimensionality related to the particle filtering technique, by introducing innovative procedures that permit a significant reduction in the number of particles. This renders the new estimator highly efficient and enables its implementation with a remarkably small number of particles. The results of a simulation study are presented, in which the new filter is compared to a recently presented unscented Kaiman filter. The comparison demonstrates the viability and robustness of the new filter and its fast convergence rate.