In this study, we analyse the $\Lambda$-$\Lambda$ hyperon channel in proton-proton scattering, $pp \mapsto \Lambda\Lambda + K^+K^+, \,\,\Lambda\mapsto p+\pi^-$, with a proton beam energy of $4.5$ GeV using the High Acceptance Di-Electron Spectrometer (HADES) at GSI.The purpose of the study presented here is to measure the production cross-section for the $\Lambda$-$\Lambda$ channel. %at the above beam Kinetic energy. The following measurement will be the first at this energy scale (CoM: $\sqrt{s}=3.46$ GeV).We develop a multi-stage analysis strategy to first determine the acceptance-reconstruction efficiency for the $\Lambda$-$\Lambda$ channel from simulations produced according to phase space.The analysis of data is perforned for three different event selections: inclusive ($ pp \mapsto \Lambda\Lambda + X$), semi-exclusive ($ pp \mapsto \Lambda\Lambda + K^+ +X$) and the fully exclusive.For each selection, the data is represented by a two-dimensional histogram, where the axes correspond to the $p+\pi^-$ invariant mass of the respective $\Lambda$ candidate. %The same strategy was employed on the experimental data to build a $\Lambda$-$\Lambda$ 2D correlation matrix. The \sloppy{$\Lambda$-$\Lambda$ signal} is determined using a Statistical sideband analysis (SSB) and a Linear least squares (LLS) fit.The inclusive selection provides the most significant $\Lambda$-$\Lambda$ signal.Cross-section values corresponding to the two analyses  are calculated as weighted average of the result of the three selections.The production cross-section for $\Lambda$-$\Lambda$ in $pp$ collisions at $\sqrt{s}=3.46$ GeV is measured \sloppy{to be $59\pm27$ nb} (one standard deviation uncertainity).