  We consider limit probabilities of first order properties in random graphs
with a given degree sequence. Under mild conditions on the degree sequence, we
show that the closure set of limit probabilities is a finite union of closed
intervals. Moreover, we characterize the degree sequences for which this
closure set is the interval $[0,1]$, a property that is intimately related with
the probability that the random graph is acyclic. As a side result, we compile
a full description of the cycle distribution of random graphs and study their
fragment (disjoint union of unicyclic components) in the subcritical regime.
Finally, we amend the proof of the existence of limit probabilities for first
order properties in random graphs with a given degree sequence; this result was
already claimed by Lynch~[IEEE LICS 2003] but his proof contained some
inaccuracies.
