Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables, $S_n=X_1+\dots+X_n$, $\Phi(x)$ be the standard normal distribution function. We investigate the asymptotics of $$ \mathbf P\{S_n>x\}/(1-\Phi(x/B_n)),\qquad n\to\infty, $$ for $0\le x\le \Lambda(B_n)$, where the function $\Lambda(z)$ is such that $$ \Lambda(z)/z\uparrow\infty,\quad\Lambda(z)/z^{1+\varepsilon}\downarrow 0\quad(0<\varepsilon<1,\ z>z_0), $$ the sequence $B_n\to\infty$ ($n\to\infty$) and $$ \sup_{x\ge 0}|\mathbf P\{S_n<xB_n\}-\Phi(x)|=o(1),\qquad n\to\infty. $$