Don't use one of those adapters that converts a 3 prong to a 2 prong unless you know for certain that the socket has been properly grounded.
You can install a new breaker in the service panel and run 3 wire Romax to your grow. Then you can install a 3 prong outlet (preferably a GFCI). Buy a basic wiring book and follow the instructions exactly. Electricians don't charge much compared to morticians, so don't be afraid to hire a pro.
Add a 20 amp, not a 15 amp. The cost is the same and the 20 amp is no more hassle than the 15 amp. While you're at it, consider installing 2 runs of wire so if your grow ever needs another breaker, the wire will already be in place.
Additional note: Adapter tab must be screwed to the outlet center screw.
Bad solution all around, as most residential outlets without ground are fed with circuits lacking ground. False sense of security can kill you.
RE: Additional circuit run. That second circuit, when run correctly, should be what is called a 3 wire circuit (Line 1, Line 2, Neutral, Ground), Can provide 240V as well as 120V, and when properly ballanced, halves the voltage drop in the wires.
Difficuly: One extra conductor.
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