Symbol timing offset (STO) and carrier frequency offset (CFO) estimation are two main synchronization operations in packet-based orthogonal frequency division multiplexing (OFDM) systems. To facilitate these operations, a periodic preamble is often placed at the beginning of a packet. CFO estimation has been extensively studied for the case of two-period preambles. In some applications, however, a preamble with more than two periods is available. A typical example is the IEEE802.11a/g wireless local area network (LAN) systems, which features a ten-period preamble. Recently, researchers have proposed a maximum likelihood (ML) CFO estimation method for such systems. This approach first estimates the received preamble using a least-squares method, and then maximizes the corresponding likelihood function. In addition to the standard calculations, this method requires an extra procedure to solve the roots of a polynomial function, which is disadvantageous for real-world implementations. In this paper, we propose a new ML method to solve the likelihood function directly and thereby perform CFO estimation. Our method can obtain a closed-form ML solution, without a need for the root-finding step. We further extend the proposed method to address the STO estimation problem, and also derive a lower bound on the estimation performance. Our simulations show that while the performance of the proposed method is either equal to or better than the existing method, the computational complexity is lower.