  A discrete countable group \Gamma is said to be ME rigid if any discrete
countable group that is measure equivalent to \Gamma is virtually isomorphic to
\Gamma. In this paper, we construct ME rigid groups by amalgamating two groups
satisfying rigidity in a sense of measure equivalence. A class of amalgamated
free products is introduced, and discrete countable groups which are measure
equivalent to a group in that class are investigated.
