Magnetic monopoles are hypothetical elementary particles exhibiting quantized magnetic charge m_0 = +/-(h/mu_0 e)) and quantized magnetic flux Phi_0 = +/-h/e. In principle, such a magnetic charge can be detected by the quantized jump in magnetic flux Phi it generates upon passage through a superconducting quantum interference device (SQUID). Naturally, with the theoretical discovery that a plasma of emergent magnetic charges should exist in several lanthanide-pyrochlore magnetic insulators, including Dy2Ti2O7, this SQUID technique was proposed for their direct detection. Experimentally, this has proven challenging because of the high number density of the monopole plasma. Recently, however, theoretical advances have allowed the spectral density of magnetic-flux noise S Phi(omega,T) due to generation recombination fluctuations of +/- m* magnetic charge pairs to be predicted. Here we report development of a SQUID based flux-noise spectrometer, and consequent measurements of the frequency and temperature dependence of S _Phi(omega,T) for Dy2Ti2O7 samples. Virtually all the elements of S _Phi(omega,T) predicted for a magnetic monopole plasma, including the existence of intense magnetization noise and its characteristic frequency and temperature dependence, are detected. Moreover, comparisons of simulated and measured correlation functions C_Phi(t) of the magnetic-flux noise Phi(t) imply that the motion of magnetic charges is strongly correlated.