The authors investigate an index transform associated with Whittaker's function \(W_{\mu,i\tau}(x)\), which has been defined by \[ [W_\mu f] (x)= \int^\infty_{-\infty} \tau\Gamma \left( \textstyle {{1\over 2}} -\mu-i\tau \right) \Gamma \left (\textstyle {1\over 2} -\mu+i \tau\right) W_{\mu,i\tau} (x)f(\tau) d \tau\;(x>0). \] This transform generalizes the familiar Kontorovich-Lebedev transform. Mapping properties and an inversion formula for this transform are established on the Lebesgue space \(L_p(\mathbb{R})\) \((p\geq 1)\). Further it is shown that the image of the transform belongs to the space \(L_{\nu,p} (\mathbb{R}_+)\) \((\nu\in \mathbb{R};1 \leq p\leq \infty)\) of functions normed by \[ \| f\|_{\nu,p} =\left( \int^\infty_0 t^{\nu p-1} \bigl| f(t) \bigr |^p dt\right)^{1/p} <\infty. \] In particular, when \(\nu=1/p\), we obtain the usual \(L_p (\mathbb{R}_+)\) space. Considering the index transform in Hilbert space, the authors also establish its Plancherel theorem.