  We define and study in detail \emph{utraslow scaled Brownian motion (USBM)\/}
characterised by a time dependent diffusion coefficient of the form $D(t)\simeq
1/t$. For unconfined motion the mean squared displacement (MSD) of USBM
exhibits an ultraslow, logarithmic growth as function of time, in contrast to
the conventional scaled Brownian motion. In an harmonic potential the MSD of
USBM does not saturate but asymptotically decays inverse-proportionally to
time, reflecting the highly non-stationary character of the process. We show
that the process is weakly non-ergodic in the sense that the time averaged MSD
does not converge to the regular MSD even at long times, and for unconfined
motion combines a linear lag time dependence with a logarithmic term. The
weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking
parameter. The USBM process is also shown to be ageing: observables of the
system depend on the time gap between initiation of the test particle and start
of the measurement of its motion. Our analytical results are shown to agree
excellently with extensive computer simulations.
