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Intermediate- and high-velocity clouds (IVCs, HVCs) are a potential source offuel for star formation in the Milky Way (MW), but their origins and fatesdepend sensitively on their distances. We search for IVC and HVC in HSThigh-resolution ultraviolet spectra of 55 halo stars at vertical heights $|z|\gtrsim 1$ kpc. We show that IVCs ($40 \leq |v_{\rm LSR}| <90$ km/s) have ahigh detection rate - the covering factor, $f_c$ - that is about constant ($f_c=0.90\pm 0.04$) from $z=1.5$ to $14$ kpc, implying IVCs are essentiallyconfined to $|z|\lesssim 1.5$ kpc. For the HVCs ($90 \leq |v_{\rm LSR}|\lesssim 170$ km/s), we find $f_c$ increases from $f_c \simeq 0.14\pm 0.10$ at$|z|\lesssim 2-3$ kpc to $f_c =0.60\pm 0.15$ at $5\lesssim |z|\lesssim 14$ kpc,the latter value being similar to that found towards QSOs. In contrast, thecovering factor of very high-velocity clouds (VHVCs, $|v_{\rm LSR}|\gtrsim 170$km/s) is $f_c<4\%$ in the stellar sample compared to 20\% in a QSO sample,implying these clouds must be at $d\gtrsim 10-15$ kpc ($|z|\gtrsim 10$ kpc).Gas clouds with $|v_{\rm LSR}|>40$ km/s at $|b|\gtrsim 15^\circ$ have therefore$|v_{\rm LSR}|$ decreasing with decreasing $|z|$. Assuming each featureoriginates from a single cloud, we derive scale-heights of $1.0 \pm 0.3$ and$2.8 \pm 0.3$ kpc for the IVCs and HVCs, respectively. Our findings providesupport to the "rain" and galactic fountain models. In the latter scenario,VHVCs may mostly serve as fuel for the MW halo. In view of their locations andhigh covering factors, IVCs and HVCs are good candidates to sustain starformation in the MW.

Intermediate- and high-velocity clouds in the Milky Way I: covering factors and vertical heights