  We study ASEP in a spatially inhomogeneous environment on a torus $
\mathbb{T}^{(N)} = \mathbb{Z}/N\mathbb{Z} $ of $ N $ sites. A given
inhomogeneity $ \widetilde{a}(x) \in (0,\infty) $, $x \in \mathbb{T} $,
perturbs the overall asymmetric jumping rates $ r < \ell \in (0,1) $ at bonds,
so that particles jump from site $ x $ to $ x+1 $ with rate $ r\widetilde{a}(x)
$ and from $x+1$ to $x$ with rate $ \ell\widetilde{a}(x) $ (subject to the
exclusion rule in both cases). Under the limit $ N\to\infty $, we suitably tune
the asymmetry $ (\ell-r) $ to zero like $N^{-\frac{1}{2}}$ and the
inhomogeneity $ \widetilde{a}(x) $ to unity, so that the two compete on equal
footing. At the level of the G\"{a}rtner (or microscopic Hopf--Cole) transform,
we show convergence to a new SPDE -- the Stochastic Heat Equation with a mix of
spatial and spacetime multiplicative noise. Equivalently, at the level of the
height function we show convergence to the Kardar--Parisi--Zhang equation with
a mix of spatial and spacetime additive noise.
  Our method applies to a general class of $ \widetilde{a}(x) $, which, in
particular, includes i.i.d., long-range correlated, and periodic
inhomogeneities. The key technical component of our analysis consists of a host
of estimates on the \emph{kernel} of the semigroup $
\mathcal{Q}(t):=e^{t\mathcal{H}} $ for a Hill-type operator $ \mathcal{H}:=
\frac12\partial_{xx} + \mathcal{A}'(x) $, and its discrete analog, where $
\mathcal{A} $ (and its discrete analog) is a generic H\"{o}lder continuous
function.
