Lipid molecules, most notably phospholipids, are the molecules that prevent cell membranes from dissolving.
A phospholipid molecule is composed of a hydrophilic head and two hydrophobic tails. In the cell membrane, phospholipid molecules are arranged in such a way that the hydrophobic tails are sandwiched between two layers of hydrophilic heads, forming a lipid bilayer. The arrangement prevents cell membranes from dissolving as only the hydrophobic heads of the molecules, which are insoluble, come in contact with water, while the hydrophilic tails, which are soluble, are shielded from exposure to water.