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The NA62 experiment reports the branching ratio measurement BR$(K^+ \rightarrow \pi^+ \nu\bar{\nu}) = (10.6^{+4.0}_{-3.4} |_{\rm stat} \pm 0.9_{\rm syst}) \times 10 ^{-11}$ at 68% CL, based on the observation of 20 signal candidates with an expected background of 7.0 events from the total data sample collected at the CERN SPS during 2016-2018. This provides evidence for the very rare decay, observed with a significance of 3.4. The experiment achieves a single event sensitivity of , corresponding to 10.0 events assuming the Standard Model branching ratio of . This measurement is also used to set limits on BR(), where is a scalar or pseudo-scalar particle. Details are given of the analysis of the 2018 data sample, which corresponds to about 80% of the total data sample.