Context :

Aims : We present new measurements of the flux densities at submillimeter wavelengths based on ALMA band 7 ( 338 GHz , \lambda 0.89 mm ) and band 9 ( 679 GHz , \lambda 0.44 mm ) observations to better constrain the origin of the continuum emission of the Mira AB binary system and to check its orbit .

Methods : We have measured the Mira A and Mira B continuum in ALMA band 7 , with a resolution of \sim 0 \aas@@fstack { \prime \prime } 31 , and for the first time in ALMA band 9 , with a resolution of \sim 0 \aas@@fstack { \prime \prime } 18 .
We have resolved the binary system at both bands , and derived the continuum spectral index of the stars and their relative position .
We also analyzed ALMA Science Verification data obtained in bands 6 and 3 .
Measurements at centimeter wavelengths obtained by other authors have been included in our study of the spectral energy distribution of the Mira components .

Results : The Mira A continuum emission has a spectral index of 1.98 \pm 0.04 extending from submillimeter down to centimeter wavelengths .
The spectral index of the Mira B continuum emission is 1.93 \pm 0.06 at wavelengths ranging from submillimeter to \sim 3.1 mm , and a shallower spectral index of 1.22 \pm 0.09 at longer wavelengths .
The high precision relative positions of the A and B components are shown to significantly depart from the current ( preliminary ) orbit by \sim 14 milliarsec .

Conclusions : The Mira A continuum emission up to submillimeter wavelengths is consistent with that of a radio photosphere surrounding the evolved star for which models predict a spectral index close to 2 .
The Mira B continuum emission can not be described with a single ionized component .
An extremely compact and dense region around the star can produce the nearly thermal continuum measured in the range \lambda 0.4 - 3.1 mm , and an inhomogeneous , less dense , and slightly larger ionized envelope could be responsible for the emission at longer wavelengths .
Our results illustrate the potential of ALMA for high precision astrometry of binary systems .
We have found a significant discrepancy between the ALMA measurements and the predicted orbit positions .