In this paper, we define a modified version τb of Kendall's tau to measure the association in a pair (X,Y ) of random variables subject to fixed left censoring due to known lower detection limits. We provide a nonparametric estimator of τb and investigate its asymptotic properties. We then assume an Archimedean copula for (X,Y ) and express τb in terms of the copula parameter α and the censoring fractions. We deduce estimators for α and for the global Kendall's tau. We develop a goodness-of-fit test for the assumed copula. We evaluate the finite-sample performance of the proposed methods by simulations and illustrate their use with a real data set on plasma and saliva viral loads. Copyright © 2011 John Wiley & Sons, Ltd.