Here's one example.  Take any ring $R$ equipped with the following two operations: $$(x,y,z)\mapsto x+y-z$$ $$(x,y,z,w)\mapsto x+(y-z)(w-x)$$

It is easy to see that if you add a constant symbol $0$, then the entire ring structure of $R$ is definable from these two operations.  However, for any $a\in R$ the map $x\mapsto x+a$ preserves these operations and thus gives an automorphism.  Hence if $R$ is any ring of cardinality $\kappa$ with no nontrivial ring-automorphisms this model will have the desired properties.  One such uncountable ring is $\mathbb{R}$; I don't know whether rigid rings of all infinite cardinalities exist but I suspect they do.

(By the way, if you're wondering how I came up with those two operations, this is just the ring analogue of a [heap](http://en.wikipedia.org/wiki/Heap_(mathematics)).  However, you can't just use heaps, because any group with more than two elements has nontrivial automorphisms.)

 Edit: As mentioned in Emil's answer, it turns out that there are rigid fields of any infinite cardinality.