DC motor vs AC motor weight considerations
Your question asks about specific power - power normalized by weight.  Generalizations of this type may be best addressed by looking at what type of motor is used in specific applications.

Electrically powered model/RC cars and planes are generally mass-constrained and optimized for performance over cost.  The present state of the art (i.e. highest performance / highest cost) uses 3 phase BLDC (brushless DC) motors with moderate torque at very high shaft speeds. Prior to the advent of compact 3 phase inverters, brush DC permanent magnet motors were used in these applications, with shaft speeds constrained by commutator limitations. One could argue that the current state of the art represents an optimal solution with regards to specific power, though such platforms are battery powered and are therefore also constrained by efficiency.  Your question did not address a usable torque range, so a very high speed motor might require gearing to work for a particular application, hurting both efficiency and specific power.

In battery-powered and hybrid automotive applications, some platforms (Prius) use 3 phase PM machines, while others (GM's truck/SUV platforms IIRC) use an induction machine.  PM machines tend to be used in smaller, higher performance applications (where performance may equate to specific power or efficiency), while induction machines tend to be used in cost-constrained applications.  

Several DARPA programs aimed at electric aircraft propulsion have used etched copper stator BLDC machines, though optimization for both specific power and efficiency is highly likely in the development/selection of such a machine.

Generalizations aside, the answer completely depends on the design of the motor.  I have seen 3 phase BLDC motors for industrial applications that are just as heavy as a comparable induction motor -- the active materials are a fraction of the total mass, which is dominated by a cast housing, conservatively-sized bearings, and environmental seals.