Unique SAT vs Exactly $m$ models
For general $m$, exactly-m-sat is strictly harder than u-sat (thus does not reduce to it) unless the PH collapses.  The reason is that PP can be obtained using an existential quantifier over exactly-m-SAT queries (exists m>(half of the assignments) such that exactly-m-SAT), thus if exactly-m-sat is in the k'th level of PH, then PP is in the (k+1)'st level, and then the hierarchy collapses (since P^PP contains PH).  But u-sat is clearly in the second level of PH (actually in a subclass called DP).

On the other hand, as @Tsuyoshi mentioned above, if $m$ is polynomial, then exactly-m-sat is many-one reudcible to u-sat.