Radio sources as weak as a few milliJanskys, such as stars and millisecond pulsars, are barely accessible to standard VLBI observations conducted with the best radio telescopes. This sensitivity limit of the VLBI technique is caused by restrictions on the maximum duration over which data may be coherently integrated due to instabilities from the atmosphere and independent clocks at the stations of an array. However, if a strong source is observed alternately with the weak source, integration times as long as in connected-element interferometry can be achieved in VLBI. The direct detections of the strong source provide reference delays and delay rates that can be used to coherently integrate the instantaneous visibilities of the weak source in each observation of a few minutes. These integrated visibilities can then be coherently added over the whole experiment by using the observed phases of the strong source as a reference. Practically, this addition is done by Fourier inversion of the integrated visibilities to produce a phase-referenced "dirty map" of the weak source yielding its right ascension and declination relative to the reference-source position. Milliarcsecond astrometry of sources as weak as a few milliJanskys is then possible. This data-processing scheme is like the phase and delay tracking of the phase center in connected-element interferometry as the Earth rotates. The stellar system Algol, in a very low state of activity with a total flux density of only 3 mJy, was observed alternately with the reference source 0309 + 411, ∼1° away, throughout a 7 hr VLBI experiment. By combining all the individual VLBI observations, the resulting coherent integration time for Algol is 4.2 hr and the corresponding signal-to-noise ratio is 12. Internal consistency checks and an analysis of possible systematic astrometric errors indicate that the accuracy of the relative VLBI position of Algol is ±0″.0005 in both coordinates and that unknown systematic effects (∼5 times the thermal noise) are presently limiting this astrometric measurement.