  A noncommutative space-time admitting dilation symmetry was briefly mentioned
in the seminal work of Doplicher, Fredenhagen and Roberts. In this paper we
explicitly construct the model in details and carry out an in-depth analysis.
The C*-algebra that describes this quantum space-time is determined, and it is
shown that it admits an action by *-automorphisms of the dilation group, along
with the expected Poincar\'e covariance. In order to study the main physical
properties of this scale-covariant model, a free scalar neutral field is
introduced as a investigation tool. Our key results are then the loss of
locality and the irreducibility, or triviality, of special field algebras
associated with regions of the ordinary Minkowski space-time. It turns out, in
the conclusions, that this analysis allows also to argue on viable ways of
constructing a full conformally covariant model for quantum space-time.
