More than four decades ago, Brochard and de Gennes proposed that colloidal suspensions of ferromagnetic particles in nematic (directionally ordered) liquid crystals could form macroscopic ferromagnetic phases at room temperature. The experimental realization of these predicted phases has hitherto proved elusive, with such systems showing enhanced paramagnetism but no spontaneous magnetization in the absence of an external magnetic field. Here we show that nanometre-sized ferromagnetic platelets suspended in a nematic liquid crystal can order ferromagnetically on quenching from the isotropic phase. Cooling in the absence of a magnetic field produces a polydomain sample exhibiting the two opposing states of magnetization, oriented parallel to the direction of nematic ordering. Cooling in the presence of a magnetic field yields a monodomain sample; magnetization can be switched by domain wall movement on reversal of the applied magnetic field. The ferromagnetic properties of this dipolar fluid are due to the interplay of the nematic elastic interaction (which depends critically on the shape of the particles) and the magnetic dipolar interaction. This ferromagnetic phase responds to very small magnetic fields and may find use in magneto-optic devices.