We know that the alternating group of degree $n>7$ has an irreducible character of degree $n-1$. The latter number is the smallest nontrivial one for each the alternating group has an irreducible character of that degree.
Does the Alternating group of degree $n>7$ have exactly one irreducible character of degree $n-1$?
Old answer: You know already that the answer is ``yes.'' For a reference, see result 2 of
Rasala, Richard On the minimal degrees of characters of $S_n$. J. Algebra 45 (1977), no. 1, 132–181.
This gives the answer for $n\geq 9$. The Atlas of Finite Groups then gives the result for $n=7,8,9$. (Note that the statement is also true for $n=7$, but not for $n=6$.)
I can email you copies of these sources should you need them.
New answer: I was unhappy with the previous answer, because one should be able to answer this using basic facts about the representation theory of the symmetric group. Here is such an answer: my reference is Fulton and Harris' Representation theory.
We know that irreducible reps of $S_n$ are associated to partitions $\lambda$ of $n$. Given such a partition $\lambda$, there are several formulae for the dimension of the associated irreducible rep - see (4.10), (4.11) and (4.12) of F&H. The latter is the hook length formula, which reads:
$$ \dim(V_\lambda)=\frac{d!}{\prod (\textrm{ Hook lengths})}.$$
Using this formula it is easy to work out that, for $n\geq 7$, there are only four reps of $S_n$ of degree less than $n$ - two of dim $1$ and two of dim $n-1$ (this is Exercise 4.14 of F&H). It's pretty obvious what they all are - each pair correspond to partitions that are conjugate to each other.
Now one uses Proposition 5.1 of F&H to see which of these remain as irreducibles when one restricts to the alternating group $A_n$ - the answer is that they remain irreducible so long as the associated partition is not self-conjugate. None of the partitions in question are self-conjugate so they remain irreducible; what is more representations corresponding to partitions that are conjugate yield isomorphic representations when one restricts to $A_n$ - this yields our irreducible reps for $A_n$ of dimension $1$ and $n-1$.
Now to complete the proof one needs to check that there are no self-conjugate partitions of $n$ for which the associated irreducible rep has dimension $\leq 2(n-1)$. (If there were these would split in half to yield irreducible representations of $A_n$.) This is easy.
As dear Derek Holt said, the answer is yes. These two references are related to this problem:
$1)$ "The Faithful Linear Representation of Least Degree of $S_n$ and $A_n$ over a Field of Characteristic 2" by $A.$ $Wagner$.
$2)$ "The Faithful Linear Representations of Least Degree of $S_n$ and $A_n$ over a Field of Odd Characteristic" by $A.$ $Wagner$.
The links for downloading these paper are:
Uniqueness.
With respect to the proposition 20.13 provided in the book Representations and characters of groups by James and Liebeck, and the answer given above by Nick Gill, the uniqueness of the "irreducible character of degree $ n−1$" is satisfied.
