Deep Keck/NIRC2 HK'L' observations of the Arches cluster near the Galactic center reveal a significant population of near-infrared excess sources. We combine the L'-band excess observations with K'-band proper motions, which allow us to confirm cluster membership of excess sources in a starburst cluster for the first time. The robust removal of field contamination provides a reliable disk fraction down to our completeness limit of H=3D19 mag, or 5 Mo at the distance of the Arches. Of the 24 identified sources with K'-L' > 2.0 mag, 21 have reliable proper motion measurements, all of which are proper motion members of the Arches cluster. VLT/SINFONI K'-band spectroscopy of three excess sources reveals strong CO bandhead emission, which we interpret as the signature of dense circumstellar disks. The detection of strong disk emission from the Arches stars is surprising in view of the high mass of the B-type main sequence host stars of the disks and the intense starburst environment. We find a disk fraction of 6 +/- 2% among B-type stars in the Arches cluster. A radial increase in the disk fraction from 3 to 10% suggests rapid disk destruction in the immediate vicinity of numerous O-type stars in the cluster core. A comparison between the Arches and other high- and low-mass star-forming regions provides strong indication that disk depletion is significantly more rapid in compact starburst clusters than in moderate star-forming environments.