Yes - there is a laundry desk in the supermarket in the end of Soi Mahatlek Luang 3 (the first side street after the BTS station / St Regis as you walk away from Siam Paragon etc). From memory it is on the ground floor of the Millenia building.

You should be able to see it on Google Maps here.

It's not cheap - they still charge per item like the hotels do, rather than per kg like the cheap places do. But the per item prices are about half the hotel rates.