The development of convective weather avoidance algorithm is crucial for aviation operations and it is also a key objective of the next generation air traffic management system. This paper proposes a novel network architecture that embeds convolutional layers into long short-time memory (LSTM) cells to predict the trajectory, based on the convective weather condition along with the flight plan before the aircraft takeoff. The data used in the experiments are history flight track data, the last on-file flight plan, and the time-dependent convective weather map. The history flight data are taken from the NASA Sherlock database and the weather data used in this paper is the Echo Top (ET) convective weather product from Corridor IntegratedWeather System (CIWS). The experiment is conducted using three months history data over the period from Nov 1, 2018 through Feb 5, 2019 with the flights from John F. Kennedy International Airport (JFK) to Los Angeles International Airport (LAX) but the methodology can be applied to the flights between any arbitrary two airports. Interpolation is performed on flight plans and real history tracks to fix the fold number of LSTM cells and also reduce computation complexity. The training loss is defined as the standard Mean Squared Error (MSE) of the predicted tracks and the real history tracks. Adam optimizer is used for backpropagation. Learning from the real historical flight data, the outof- sample test shows that 47.0% of the predicted flight tracks can reduce the deviation compared to the last on-file flight plan. The overall variance is reduced by 12.3%.