The answer is positive. The reason is that exceptions hide the types of values that they communicate, which may be recursive types.

We first to define two inverse functions fold and unfold, using the following exception to mask types of values:

exception Hide of ((unit -> unit) -> (unit -> unit));;
let fold f = fun (x : unit) -> (raise (Hide f); ()) let unfold f = try (f(): unit); fun x -> x with Hide y -> y;;
val fold : ((unit -> unit) -> unit -> unit) -> unit -> unit = <fun> val unfold : (unit -> unit) -> (unit -> unit) -> unit -> unit = <fun>

The two functions fold and unfold are inverse coercions between type UU and U where U is unit → unit: They can be used to embed any term of the untyped lambda calculus into a well-typed term, using the following well-known encoding:

[[x]]= x
[[λ x. a]]= foldx. [[a]])
[[a1 a2]]= unfold ([[a1]] [[a2]])

In particular, [[fix ]] is well-typed.