  [ABRIDGED] The Cash statistic, also known as the C stat, is commonly used for
the analysis of low-count Poisson data, including data with null counts for
certain values of the independent variable. The use of this statistic is
especially attractive for low-count data that cannot be combined, or re-binned,
without loss of resolution. This paper presents a new maximum-likelihood
solution for the best-fit parameters of a linear model using the Poisson-based
Cash statistic. The solution presented in this paper provides a new and simple
method to measure the best-fit parameters of a linear model for any
Poisson-based data, including data with null counts. In particular, the method
enforces the requirement that the best-fit linear model be non-negative
throughout the support of the independent variable. The method is summarized in
a simple algorithm to fit Poisson counting data of any size and counting rate
with a linear model, by-passing entirely the use of the traditional $\chi^2$
statistic.
