The sentence is perfectly grammatical as it is. Here a friend of mine's is called a phrasal genitive. We say that mine's has double case-marking, an inner case and an outer case. See especially the discussion following the examples in , below.
16.6 Head and phrasal genitives
Genitive NPs [noun phrases] are usually marked as such by the inflection of the head noun: we refer to these as head genitives. It is also possible for the marking to be located on the last word of a (final) post-head dependent, and these we call phrasal genitives. Compare, then, the following, where underlining [boldface] marks the head and the genitive suffix:
 HEAD GENITIVE PHRASAL GENITIVE
i a. [Edward's] daughter b. [ the King of England 's] daughter
ii a. [everyone's] responsibility b. [everyone else 's] responsibility
iii a. [somebody's] initiative b. [somebody local 's] initiative
iv a. [ the doctor's] house b. [ a guy I know 's] house
In [ib], for example, the head of the genitive NP is King but the suffix 's attaches to England, the last word of the phrase the King of England. The range of phrasal genitive constructions is greater in informal, especially spoken, styles than in formal and written ones. In the latter it is normally restricted to post-head dependents with the form of a PP, including else, as in [ib/iib]. In informal speech it is also found with relative clauses ([ivb]), and the occasional postpositive AdjP ([iiib]). Acceptability decreases as the weight or complexity of the post-head dependent increases, as illustrated in the following examples:
 i a. [ the Head of Department 's] speech
b ?[ the Head of the newly formed Asian Studies Department 's] speech
ii a. [ the man she was speaking to 's] reaction
b. ?[ the man she and her friend had been complaining to 's] reaction
c. #[the man she and her friend had been complaining to so angrily 's] reaction
Example [ia] is acceptable in any style, and [iia] is fully acceptable in informal speech, but the rest are marginal and would generally be avoided in favour of other constructions, notably the + N + PP (the speech by the Head of the newly formed Asian Studies Department). And in [ii] version [c] can be regarded as significantly worse than [b]: examples of this degree of complexity are certainly unacceptable.
■ Why we need to distinguish the head and phrasal genitive constructions
In both head and phrasal genitives the genitive marking is on the last word. It is not possible to have genitive marking on the head if there is a post-head dependent: we cannot have *[ the Head's of Department] speech or *[ the man's she was speaking to] reaction.65
65Exceptions to this rule are occasionally found in the Type III construction (i.e. with a fused determiner-head): I could feel the hair stand up on the back of my neck like [a dog's that is going to get into a fight]. The relative clause is too long to accept the 's marker at the end, but one would have expected this to lead to the avoidance of the genitive: like that of a dog that is going to get into a fight. Examples of the kind cited are not acceptable and frequent enough to qualify as grammatical.
Why then, it might be asked, do we need to distinguish between head and phrasal genitives — can't we treat all genitives as phrasal genitives (i.e. marked on the final word), with examples like Edward's daughter, the manager's departure, etc., simply having the head as the final word? The reason why this would not do, why we need to distinguish head and phrasal genitives, is that the morphological realisation of genitive case is sensitive to the distinction. This can be seen in such pairs as:
 i a. [my] facial expression b. [the man opposite me's] facial expression
ii a. [my friend's] father b. [a friend of mine's] father
In [i] the genitive is marked on the pronoun I: in the head genitive construction [a] the pronoun is realised as my but in the phrasal genitive as me's. In [ii] we have two genitives: in [iia] one is again realised as my and the other as friend's, whereas in [iib] they combine in the single word mine's. Both me's and mine's thus have double case-marking, an inner case and an outer case. In me's the inner case is accusative, required because the pronoun is object of the preposition opposite, while in [iib] the inner case is genitive because the pronoun is functioning in the oblique genitive construction. The phrasal genitive is the outer case, morphologically added to the form that realises the inner case. In examples like [ the King of England's] daughter the inner case of England's is the plain case, which has no morphological marking, but the principle is the same: the outer genitive is added to the form required by England within the inner NP the King of England. In the man she was speaking to's we only have an outer layer of case because the last word of the inner NP, to, is a preposition, not a noun, and hence does not have inner case.