Cuprates and other high-temperature superconductors consist of two-dimensional layers that are crucial to their properties. The dynamics of the quantum spins in these layers lie at the heart of the mystery of the cuprates(1-7). In bulk cuprates such as La2CuO4, the presence of a weak coupling between the two-dimensional layers stabilizes a three-dimensional magnetic order up to high temperatures. In a truly two-dimensional system however, thermal spin fluctuations melt long-range order at any finite temperature(8). Here, we measure the spin response of isolated layers of La2CuO4 that are only one-unit-cell-thick. We show that coherent magnetic excitations, magnons, known from the bulk order, persist even in a single layer of La2CuO4, with no evidence for more complex correlations such as resonating valence bond correlations(9-11). These magnons are, therefore, well described by spin-wave theory (SWT). On the other hand, we also observe a high-energy magnetic continuum in the isotropic magnetic response that is not well described by two-magnon SWT, or indeed any existing theories.