The launching mechanism of the jets of active galactic nuclei is poorly constrained observationally, owing to the large distances to these objects and the very small scales (sub-parsec) involved. To better constrain theoretical models, it is especially important to get information from the region close to the physical base of the jet, where the plasma acceleration takes place. In this paper, we report multi-epoch and multifrequency continuum observations of the z = 2.5 blazar PKS 1830−211 with ALMA, serendipitously coincident with a strong γ-ray flare reported by Fermi-LAT. The blazar is lensed by a foreground z = 0.89 galaxy, with two bright images of the compact core separated by 1 . Our ALMA observations individually resolve these two images (although not any of their substructures), and we study the change in their relative flux ratio with time (four epochs spread over nearly three times the time delay between the two lensed images) and frequency (between 350 and 1050 GHz, rest frame of the blazar), during the γ-ray flare. In particular, we detect a remarkable frequency-dependent behavior of the flux ratio, which implies the presence of a chromatic structure in the blazar (i.e., a core-shift effect). We rule out the possibility of micro-and milli-lensing effects and propose instead a simple model of plasmon ejection in the blazar's jet to explain the time and frequency variability of the flux ratio. We suggest that PKS 1830−211 is likely to be one of the best sources to probe the activity at the base of a blazar's jet at submillimeter wavelengths, thanks to the peculiar geometry of the system. The implications of the core shift in absorption studies of the foreground z = 0.89 galaxy (e.g., constraints on the cosmological variations of fundamental constants) are discussed.