We present ALMA observations of the 2M1207 system, a young binary made of a brown dwarf with a planetarymass companion at a projected separation of about 40 au. We detect emission from dust continuum at 0.89 mm and from the J 3 2 = -rotational transition of CO from a very compact disk around the young brown dwarf. The small radius found for this brown dwarf disk may be due to truncation from the tidal interaction with the planetarymass companion. Under the assumption of optically thin dust emission, we estimatea dust mass of 0.1 M ⊕ for the 2M1207A diskand a 3σ upper limit of ∼1 M Moon for dust surrounding 2M1207b, which is the tightest upper limit obtained so far for the mass of dust particles surrounding a young planetary-mass companion. We discuss the impact of this and other non-detections of young planetary-mass companions for models of planet formation thatpredictcircumplanetary material tosurroundthese objects.