  Consider a multidimensional diffusion process $X=\{X\left(t\right)
:t\in\lbrack0,1]\}$. Let $\varepsilon>0$ be a \textit{deterministic}, user
defined, tolerance error parameter. Under standard regularity conditions on the
drift and diffusion coefficients of $X$, we construct a probability space,
supporting both $X$ and an explicit, piecewise constant, fully simulatable
process $X_{\varepsilon}$ such that \[ \sup_{0\leq t\leq1}\left\Vert
X_{\varepsilon}\left(t\right) -X\left(t\right) \right\Vert_{\infty}<\varepsilon
\] with probability one. Moreover, the user can adaptively choose
$\varepsilon^{\prime}\in\left(0,\varepsilon\right) $ so that
$X_{\varepsilon^{\prime}}$ (also piecewise constant and fully simulatable) can
be constructed conditional on $X_{\varepsilon}$ to ensure an error smaller than
$\varepsilon^{\prime}>0$ with probability one. Our construction requires a
detailed study of continuity estimates of the Ito map using Lyon's theory of
rough paths. We approximate the underlying Brownian motion, jointly with the
L\'{e}vy areas with a deterministic $\varepsilon$ error in the underlying rough
path metric.
