'It did not always seem of the same size or weight' (FR 1.ii.47)

'Though [Bilbo] had found out that the [Ring] needed looking after; it did not seem always of the same size or weight; it shrank or expanded in an odd way, and might suddenly slip off a finger where it had been tight.'

(FR 1.ii.47) 

To argue that the Ring does not in truth change weight is fairly easy. For, although both Frodo and Sam feel it to be a physically heavy burden while they are bearing it, Sam is surprised to find when he carries Frodo up Mt Doom that he feels only the weight of Frodo (RK 6.iii.940). The changes in size, however, are probably not illusory. 

Recall that Sauron changes shape and appearance more than once in The Silmarillion, from human to werewolf to vampire bat (175). Recall also that in the Second Age, during which he forges the One Ring and transfers much of his native power into it, he can still change his form and appearance (285). After he is killed in the Drowning of Númenor, however, he can no longer 'put on his fair hue' (285) or 'appear fair to the eyes of Men' (280); and after he is killed again and the Ring taken from him he cannot take shape again for centuries (UT 388-89). Now since he had been able to change his size and shape previously it makes sense that the Ring would need to possess the same capacity if only for practical purposes. After his deaths the power he put into the Ring remains in the Ring, and so it adjusts to the hand that wears it.