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We report measurements of the Meissner screening profile in a Nb(300 nm)/Al$_{2}$O$_{3}$ thin film using $^{8}$Li $\beta$-detected nuclear magnetic resonance ($\beta$-NMR). The NMR probe $^{8}$Li was ion-implanted into the Nb film at energies $\leq$ 20 keV, corresponding to mean stopping depths comparable to Nb's magnetic penetration depth $\lambda$. $^{8}$Li's strong dipole-dipole coupling with the host $^{93}$Nb nuclei provided a "cross-relaxation" channel that dominated in low magnetic fields, which conferred indirect sensitivity to the local magnetic field via the spin-lattice relaxation (SLR) rate $1/T_{1}$. From a fit of the $1/T_{1}$ data to a model accounting for its dependence on temperature, magnetic field, and $^{8}$Li$^{+}$ implantation energy, we obtained a magnetic penetration depth $\lambda_{0}$ = 51.5(22) nm, consistent with a relatively short carrier mean-free-path $\ell$ = 18.7(29) nm typical of similarly prepared Nb films. The results presented here constitute an important step towards using $^{8}$Li $\beta$-NMR to characterize bulk Nb samples with engineered surfaces, which are often used in the fabrication of particle accelerators.