Hydrodynamical simulations show that a pair of spiral arms can form in the disk around a rapidly growing young star and that the arms are crucial in transporting angular momentum as the disk accretes material from the surrounding envelope1-4. Here we report the detection of a pair of symmetric spiral structures in a protostellar disk, supporting the formation of spiral arms in the disk around a forming star. The HH 111 VLA 1 source is a young Class I source embedded in a massive infalling protostellar envelope and is actively accreting, driving the prominent HH 111 jet. Previous observations showed a ring of shock emission around the disk's outer edge5, indicating accretion of the envelope material onto the disk at a high rate. Now with ALMA observations of thermal emission from dust particles, we detect a pair of spiral arms extending from the inner region to the disk's outer edge, similar to that seen in many simulations1-4. Additionally, the disk is massive, with a Toomre Q parameter near unity in the outer parts where the spiral structures are detected, supporting the notion that envelope accretion is making the outer disk gravitationally unstable. In our observations, another source, HH 111 VLA 2, is spatially resolved for the first time, showing a disk-like structure with a diameter of 26 au and an orientation nearly orthogonal to that of the HH 111 VLA 1 disk.