This paper develops a method for obtaining confidence bands in polynomial regression when the observations are independently distributed with constant but unknown variance. The bands may be obtained, in principle, over arbitrary sets of the independent variable with exact preassigned confidence coefficients. In general, difficult distribution problems result when specific applications are attempted. The major portion of this paper is concerned with first degree polynomials since some progress has been made here. A table is provided to obtain a constant width confidence band which contains the true but unknown straight regression line for values of the independent variable in some arbitrarily selected interval with an exact preassigned confidence coefficient. The present method is compared with the classical hyperbolic band for the whole regression line.