In this paper, a novel approach is proposed for estimating traffic matrices. Our method, called PamTram for PArtial Measurement of TRAffic Matrices, couples lightweight origin-destination (OD) flow measurements along with a computationally lightweight algorithm for producing OD estimates. The first key aspect of our method is to actively select a small number of informative OD flows to measure in each estimation interval. To avoid the heavy computation of optimal selection, we use intuition from game theory to develop randomized selection rules, with the goals of reducing errors and adapting to traffic changes. We show that it is sufficient to measure only one flow per measurement period to drastically reduce errors-thus rendering our method lightweight in terms of measurement overhead. The second key aspect is an explanation and proof that an Iterative Proportional Fitting algorithm approximates traffic matrix estimates when the goal is a minimum mean-squared error; this makes our method lightweight in terms of computation overhead. A one-step error bound is provided for PamTram that bounds the average error for the worst scenario. We validate our method using data from Sprint's European Tier-1 IP backbone network and demonstrate its consistent improvement over previous methods.