Orbit determination for Near-Earth Asteroids presents unique technical challenges due to the imperative of early detection and careful assessment of the risk posed by specific Earth close approaches. This article presents a case study of asteroid 99942 Apophis, a 300-400 meter object that, for a short period in December 2004, held an impact probability of more than 2% in 2029. Now, with an orbit based on radar ranging and more than a year of optical observations, we can confidently say that it will pass safely by the Earth in 2029, although at a distance of only about six Earth radii from the geocenter. However, the extremely close nature of this encounter acts to obscure the trajectory in subsequent years, when resonant returns to the vicinity of the Earth are possible. In particular, an impact possibility in the year 2036 has a roughly 5% probability of persisting through the very favorable 2013 radar and optical observing apparition. In the event that the 2036 potential impact has not been eliminated by 2013, a precise characterization of the Yarkovsky accelerations acting on the asteroid may become an important part of the orbit estimation and impact prediction problem. Even so, the sixteen years available to effect a deflection from 2013 until 2029, after which the problem would become intractable, are sufficient to respond to the threat should a deflection effort become warranted.