The efficiency (measured in terms of ratio of average sample size to fixed sample size) of the Wald sequential probability ratio test relative to the best competing fixed sample procedure for testingH0:θ = θ0versusH1:θ = θ1(θ0< θ1) whenXis distributedN(X|θ, σ2) with known σ1is studied for fixed (θ, θ0, θ1) as the specified probabilities of error, α (the probability of rejectingH0when θ = θ0) and β (the probability of acceptingH0when θ = θ1), approach zero. The limit is shown to depend on the relative rates at which α and β approach zero in a prescribed manner. In particular, whenaα= β(a> 0) this limiting relative efficiency is equal to (θ1– θ0)/(4|θ1+θ0− 2θ|). The practical implications of this result are discussed.