According to wikipedia (Sept. 2011), the algorithm was first published by Kenneth Levenberg, while working at the Frankford Army Arsenal. It was rediscovered by Donald Marquardt who worked as a statistician at DuPont and independently by Girard, Wynn and Morrison.

K Levenberg (1944). A Method for the Solution of Certain Non-Linear Problems in Least Squares. The Quarterly of Applied Mathematics 2: 164–168.

DW Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters, J. Society for Industrial and Applied Mathematics, 11:431-441, 1963.