A Caribbean mystery : a Miss Marple mystery / Agatha Christie.

Enhanced descriptions from Syndetics:

The tiny Caribbean island of St. Honore is a tropical paradise. But for Miss Marple, enjoying a well-earned rest from her busy life in the village of St. Mary Mead, it is a place where nothing ever seems to happen. Until old Major Palgrave tells her the strange story of a suspected double murderer. Eventually, her interest is aroused by an old soldier's yarn about a murderer he had known. Just as Major Palgrave is about to show the murderer's picture to her, he looks over her shoulder, appears startled, and changes the subject. The next morning, a servant finds him dead in his room. Doctor Graham concludes that the major died of heart failure, since he showed all the symptoms, and had a bottle of serenite on his table. As rumors begin to circulate, Miss Marple is not alone in suspecting that things are not as they seem. And when a death that is indisputably murder occurs, she finds an unlikely ally in the cantankerous crippled millionaire Mr. Rafiel.