Consider a linear regression model with n-dimensional response vector, p-dimensional regression parameter β and independent normally distributed errors. Suppose that the parameter of interest is θ = a T β where a is a specified vector. Define the s-dimensional parameter vector τ = C T β − t where C and t are specified. Also suppose that we have uncertain prior information that τ = 0. Part of our evaluation of a frequentist confidence interval for θ is the ratio (expected length of this confidence interval)/(expected length of standard 1 − α confidence interval), which we call the scaled expected length of this interval. We say that a 1 −α confidence interval for θ utilizes this uncertain prior information if (a) the scaled expected length of this interval is significantly less than 1 when τ = 0, (b) the maximum value of the scaled expected length is not too large and (c) this confidence interval reverts to the standard 1 − α confidence interval when the data happen to strongly contradict the prior information. LetΘ = a Tβ andτ = C Tβ − t, whereβ is the least squares estimator of β. We consider the particular case that that E (τ − τ )(Θ − θ) = 0, so thatΘ andτ are independent. We present a new 1 − α confidence interval for θ that utilizes the uncertain prior information that τ = 0. The following problem is used to illustrate the application of this new confidence interval. Consider a 2 3 factorial experiment with 1 replicate. Suppose that the parameter of interest θ is a specified linear combination of the main effects. Assume that the three-factor interaction is zero. Also suppose that we have uncertain prior information that all of the two-factor interactions are zero. Our aim is to find a frequentist 0.95 confidence interval for θ that utilizes this uncertain prior information.