Summary: Let \(\mathcal P\) be a Poisson process of intensity 1 in a square \(S_n\) of area \(n\). For a fixed integer \(k\), join every point of \(\mathcal P\) to its \(k\) nearest neighbours, creating an undirected random geometric graph \(G_{n,k}\). We prove that there exists a critical constant \(c_{\text{crit}}\) such that, for \(c< c_{\text{crit}}\), \(G_{n,\lfloor c\log n\rfloor} \) is disconnected with probability tending to 1 as \(n \rightarrow \infty \) and, for \(c> c_{\text{crit}}\), \(G_{n,\lfloor c\log n\rfloor} \) is connected with probability tending to 1 as \(n \rightarrow \infty \). This answers a question posed in [P. Balister, B. Bollobás, A. Sarkar, and M. Walters, Adv. Appl. Probab. 37, No.1, 1–24 (2005; Zbl 1079.05086)].