Martin Hoecker-Martinez's Answer is correct for perfectly noiseless observations of a two body Kepler system, i.e. the force between the bodies is directed along the vector linking them and the force magnitude follows and inverse square law with distance. An alternative to Martin's answer is that perfectly known position and velocity will determine all future motions for the two body problem.
However, these assumptions do not strictly hold: a comet not only interacts with the Sun but also the planets (especially Jupiter) and other gravitational sources and moreover there are other nonideal "noises" that one must account for (solar wind and so forth) as in CuriousOne's answer.
Practically, the way one handles this is to assume a Kepler model (or more elaborate model if you know where all the planets are and can thus account for their effects) and treat other perturbations as additive Gaussian noise. One then finds the current maximum likelihood estimate of the orbit model parameters (position, velocity) using a Kalman Filter Algorithm, which I describe here and here. So you will begin six observations and then make new ones regularly, using the Kalman filter to update your estimates. The Kalman filter will give you current variances on your estimates so you can always put fairly rigorous error bounds on theoretically calculated future positions. So there will be some finite number of observations needed for whatever accuracy you need in your calculations: the Kalman filter will let you know when you have enough.
Historically, exactly your problem motivated the invention of the Kalman filter. For, although we credit Rudolf Kalman as its inventor, it was in fact first published in 1809 by Carl Friedrich Gauss, where he documented his use of it for simplifying hand calculations made in estimating the orbital parameters of celestial bodies. See
"Recursive Estimation and the Kalman Filter" in D.G.S. Pollock's "Kalman Filters"