A beverage maker has a valve structure (13; 43; 153) adapted block the return of water from a boiler (5) to its water reservoir (2) in response to a water pressure from the boiler exceeding a shut-off pressure value. The valve structure (7; 47; 107) further includes a discharge outlet (20; 120) connected for water outflow to the brewing unit (8) and is adapted to open a passage from that inlet (9; 109) to the discharge outlet (20; 120) in response to a water pressure at the inlet (9; 109) exceeding a discharge pressure value below the pumping pressure. The valve structure (7; 47; 107) is further adapted to block the return of water from the boiler (5) to the water reservoir (2) also in response to a water pressure from the boiler below a relief pressure value above atmospheric pressure and below the shut-off pressure value. The pressure in the boiler during heating towards boiling is thus kept above atmospheric pressure.