Let $X_1$, $X_2$, ${cdots}$, $X_n$ be independent and identically distributed random variables having common exponential density with unknown mean ${mu}$. In the sequential confidence interval estimation for the exponential hazard rate ${ heta}=1/{mu}$, when the loss function is strictly convex, the following stopping rule is proposed with the half length d of prescribed confidence interval $I_n$ for the parameter ${ heta}$; ${ au}$ = smallest integer n such that $n{geq}z^2_{{alpha}/2}hat{ heta}^2/d^2+2$, where $hat{ heta}=(n-1)ar{X}{_n}^{-1}/n$ is the minimum risk estimator for ${ heta}$ and $z_{{alpha}/2}$ is defined by $P({mid}Z{mid}{leq}{alpha}/2)=1-{alpha}({alpha}{in}(0,1))$ Z ~ N(0, 1). For the confidence intervals $I_n$ which is required to satisfy $P({ heta}{in}I_n){geq}1-{alpha}$. These estimated intervals $I_{ au}$ have the asymptotic consistency of the sequential procedure; $$lim_{d{ ightarrow}0}P({ heta}{in}I_{ au})=1-{alpha}$$, where ${alpha}{in}(0,1)$ is given.