  We analyze the probability of ruin for the {\it scaled} classical
Cram\'er-Lundberg (CL) risk process and the corresponding diffusion
approximation. The scaling, introduced by Iglehart \cite{I1969} to the
actuarial literature, amounts to multiplying the Poisson rate $\la$ by $n$,
dividing the claim severity by $\sqrtn$, and adjusting the premium rate so that
net premium income remains constant. %Therefore, we think of the associated
diffusion approximation as being "asymptotic for large values of $\la$."
  We are the first to use a comparison method to prove convergence of the
probability of ruin for the scaled CL process and to derive the rate of
convergence. Specifically, we prove a comparison lemma for the corresponding
integro-differential equation and use this comparison lemma to prove that the
probability of ruin for the scaled CL process converges to the probability of
ruin for the limiting diffusion process. Moreover, we show that the rate of
convergence for the ruin probability is of order $\mO\big(n^{-1/2}\big)$, and
we show that the convergence is {\it uniform} with respect to the surplus. To
the best of our knowledge, this is the first rate of convergence achieved for
these ruin probabilities, and we show that it is the tightest one in the
general case. For the case of exponentially-distributed claims, we are able to
improve the approximation arising from the diffusion, attaining a uniform
$\mO\big(n^{-k/2}\big)$ rate of convergence for arbitrary $k \in \N$. We also
include two examples that illustrate our results.
