On the accuracy of the normal approximation. I

Presented are practically applicable estimates of the accuracy of the normal approximation for the distributions of sums of independent identically distributed absolutely continuous random variables with finite moments of order $2+\delta$, $0<\delta\le 1$. The right-hand side of the estimate is the sum of two summands, the first being the Lyapunov fraction with the absolute constant arbitrarily close to the asymptotically exact one, whereas the second summand decreases exponentially fast.