  Let $(W,W')$ be an exchangeable pair. Assume that \[E(W-W'|W)=g(W)+r(W),\]
where $g(W)$ is a dominated term and $r(W)$ is negligible. Let
$G(t)=\int_0^tg(s)\,ds$ and define $p(t)=c_1e^{-c_0G(t)}$, where $c_0$ is a
properly chosen constant and $c_1=1/\int_{-\infty}^{\infty}e^{-c_0G(t)}\,dt$.
Let $Y$ be a random variable with the probability density function $p$. It is
proved that $W$ converges to $Y$ in distribution when the conditional second
moment of $(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen
type bound is also given. We use this technique to obtain a Berry-Esseen error
bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the
magnetization in the Curie-Weiss ferromagnet at the critical temperature.
Exponential approximation with application to the spectrum of the
Bernoulli-Laplace Markov chain is also discussed.
